Proceedings of the Combustion Institute, Volume 29, 2002/pp. 1873–1879
EFFECTS OF FLAME STRETCH AND WRINKLING ON CO FORMATION IN TURBULENT PREMIXED COMBUSTION P. NILSSON and X. S. BAI Division of Fluid Mechanics Lund University S 221 00 Lund, Sweden
This paper presents an investigation of CO formation in a lean premixed propane/air turbulent flame in an afterburner configuration. A previous experiment showed that a high amount of CO was formed in the mean turbulent flame brush. To explain the high CO concentration in the flame zone, the effects of flame stretch and flame wrinkling are studied, based on an ensemble averaged laminar flamelet library approach. It is shown that the flame stretch decreases the laminar burning velocity by 20% under the studied flame conditions, and the stretched flamelet model predicts the non-equilibrium CO concentration in the postflame zone. However, the high CO concentration in the mean flame brush cannot be predicted by a stretched flamelet library model alone. A flamelet model accounting for wrinkled flamelets in the mean turbulent flame brush, and the effect of flame stretch (mainly strain rate), is tested. The model is based on a level-set G-equation for the mean position of the turbulent flame brush and an ensemble average of strained laminar flamelet libraries. A comparison of the numerical results with the experimental data and a previous translating flamelet model clearly shows that the wrinkled flamelet model predicts the intermediate species, such as CO, more accurately. The major species such as O2 and CO2, as well as temperature, are found to be not sensitive to the flame wrinkling.
Introduction Lean premixed turbulent combustion has the advantage of low emissions of pollutants such as NOx and has great potential for application in the gas turbine industry. To understand the fundamental process of lean premixed turbulent combustion, experimental and computational investigations have been reported (cf. Ref. [1] and references therein). An essential characteristic of premixed turbulent combustion is flame-eddy interaction. The interaction of turbulence eddies and flame affects the multiplelayer structure of the flame. Asymptotic analyses of hydrocarbon/air flame structures [2,3] indicate that competition among the elementary reaction rates, mass and heat transfer rates in the multiple thin layers, influences the formation of intermediate species and flame burning velocity. The flame stretch, a combination effect of aerodynamic strain rate and flame curvature, particularly with differential diffusion of heat and chemical species, modifies the burning velocity and may even cause local flame extinction. Numerous studies on the response of the flame burning velocity and extinction to the flame stretch have been carried out [4–11]. In the limit of large activation energy and low stretch, the flame burning velocity was shown to be linearly dependent on the flame stretch [4–6]. These theoretical analyses provide insight to the physical process and set up a basis for developing robust models for numerical simulation of turbulent premixed combustion.
For low-intensity turbulence and thin flames (i.e., the flame is thinner than the Kolmogorov microscale), the instantaneous flames may be approximated as laminar flamelets [12]. As a first approximation, the thin laminar flamelet may be modeled as an infinitely thin surface fluctuating in the turbulent flowfield, and the mean turbulent flame may be modeled based on a presumed probability density function (PDF), for example using the Bray-LibbyMoss formulation [13]. Models accounting for increased flame surface area per volume in the flowfield, as a result of flame wrinkling, have been developed [14]. The flame surface density concept has been used in recent large eddy simulations and Reynolds averaged Navier-Stokes (RANS) equation simulations to model spatial filtered or Reynolds averaged mean reaction rates [15]. Stretched flamelet structures of finite thickness have been explicitly employed to model the turbulent mean flame using a flamelet library approach [16], in which the mean flame position in turbulent premixed combustion is modeled using the level-set G-equation; the mean flame properties such as density, species mole fraction, and temperature are computed using an ensemble averaged flamelet library based on presumed PDFs. The present work is aimed at investigation of the effects of flame stretch and flame wrinkling on the mean turbulent flame, particularly the formation of CO in lean premixed combustion. An experimental
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TURBULENT COMBUSTION—Measurements in Premixed Turbulent Flames
Fig. 1. A sketch of the afterburner (experimental rig) configuration, the mean flame brush, the wrinkled flamelets, and the flow pattern. Width of the combustor: 0.12 m in the y-direction, 0.24 m in the z-direction (perpendicular to the shown x-y plane). Size of the triangular prism-shaped flame holder: 0.04 m in the y-direction, 0.24 m in the zdirection.
flame [17] with premixed propane/air is chosen. The flame, with equivalence ratio 0.6 and a preheat temperature of 600 K, is stabilized using a bluff body in an afterburner configuration. The CO field in such lean combustion is particularly difficult to model, since the production and formation of CO occur only in very thin layers, and the net formation of CO depends strongly on the flame stretch and flow residence time. It is therefore very sensitive to the model parameters. Previous simulations using the flamelet library approach [18] or the eddy dissipation model [17] showed that the calculated CO mole fraction was considerably lower than the measured value. In the present work, a model accounting for the stretched flamelet structures and the increased flame surface area due to wrinkling, proposed by Peters [16], is employed. The response of flamelet structures to the flame stretch (mainly strain rate) in a ‘‘back-to-back’’ twin flame configuration is studied using a detailed chemical kinetic mechanism [19]. It is shown that the stretched flamelet model is useful for studying the CO emissions in the postflame zones, but fails to explain the high CO concentration found in the mean flame zone in the experiments. The flame wrinkling accounts for the high CO concentration in the flame zone. It is expected that other intermediate species, such as radicals, should also be most sensitive to the modeling of flame wrinkling in the mean flame zones. Response of Flamelet Structures to Flame Stretch Consider the afterburner configuration shown in Fig. 1. First, the influence of flame stretch on the local laminar flamelet structure is examined. The flame surface in turbulent flows is a complex threedimensional configuration. In the flamelet regime,
the flame thickness may be assumed much thinner than the radius of the curvature. A flamelet equation with flame stretch as an input parameter can be employed for the calculation of the stretched flamelet libraries [20]. The flamelet structure under pure curvature may be modeled in a spherical flame configuration [21]. In the current application, the stretch flame is modeled using a back-to-back counterflow twinflame configuration and, alternatively, a ‘‘fresh-toburned’’ counterflow configuration. These flames have been studied in numerical and asymptotic analyses by Rogg and Peters [8,9]. The stretch rate, which is identical to the strain rate in the current configuration, is defined as K ⳱ d(ln A)/dt. For different strain rates, from zero strain (unstretched flame) to quenching strain rate, calculations using an in-house code [22], with detailed chemistry consisting of 30 species and 82 reactions [19], are performed. The data are stored in a two-dimensional (flamelet distance function G and strain rate K) flamelet library, denoted as Yi(G, K), q(G, K), T(G, K), for mass fractions of species i, density, and temperature of the mixture, respectively. Figure 2 shows the influence of the aerodynamic strain rate on the laminar burning velocity. To exclude ambiguity, the flame surface is defined as the location where the intermediate species CH2O has its peak concentration; sL is defined as (qv)0/qu, where v is the flow velocity component along the incoming flow direction. The subscript 0 denotes the location of G ⳱ 0, and u denotes the unburned mixture. In the figure, s0L denotes the unstretched laminar flame burning velocity, which is 0.743 m/s, obtained from a numerical calculation on an 1.2
1.0 backtoback freshtoburned
0.8 SL/SL0
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0.6
0.4
0.2
0.0
0
500
1000
1500
2000
2500
3000
Strain rate [1/s] Fig. 2. Burning velocity at different strain rates, calculated using a detailed chemical kinetic mechanism [18] in a back-to-back and a fresh-to-burned counterflow configuration.
CO FORMATION IN PREMIXED TURBULENT COMBUSTION
O2
CO
0.2
O2 mole fraction
0.04
1
K=0 s 1 K=500 s 1 K=1000 s 1 K=2200 s
0.03
0.15
0.02
0.1
0.01
0.05 0.5
0.25
0
0.25
0.5
0.75
1
CO mole fraction
0.25
0
G / mm
Fig. 3. Distributions of O2 and CO in the flamelet coordinate, at different strain rates, calculated using a detailed chemical kinetic mechanism [18] in a back-to-back, counterflow configuration.
one-dimensional free propagating flame configuration, with the same chemistry and transport properties as in the stretched flame calculations. As seen, at low strain rates, the burning velocities calculated using the two counterflow configurations agree with each other, and they are approximately linearly dependent on the strain rate. This result confirms the analysis of Ref. [5] and the numerical and asymptotic results of Refs. [8,9]. As the strain rate increases, the laminar burning velocity decreases. The figure indicates that for moderately low strain rate flames (strain rate ⬍ 1000 sⳮ1), both configurations may be used to generate the laminar flamelet library. At high strain rate, the burning velocity calculated from the two configurations is significantly different. At about 2200 sⳮ1, the slope of the sL versus K profile calculated from the back-to-back configuration tends to negative infinity. Increasing the strain rate further leads to flame extinction. The fresh-toburned configuration predicts a much later flame extinction (at a strain rate of about 5000 sⳮ1). The flame extinction can be further understood from the CO and O2 profiles shown in Fig. 3. As seen, at the inner layer a high superequilibrium CO peak is found. At large positive G (postflame zone), a low CO concentration is found. In the case of zero strain rate, the CO mole fraction at a far downstream point in the postflame zone (G ⳱ 20 mm) reaches its equilibrium, which is very low (about 180 ppm in mole fraction) in this lean flame. Increasing the strain rate leads to a decrease in the flow residence time; therefore, the CO mole fraction at the postflame zone cannot reach its equilibrium. Instead, a rather high CO is found, indicating incomplete combustion. For a strain rate of 500 sⳮ1 the corresponding CO mole fraction is about 1200 ppm far downstream in the postflame zone, almost an order of magnitude higher than the unstretched value. It is
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interesting to note that the superequilibrium CO peak in the inner layer is only slightly affected by the strain rate. The peak value of CO decreases by about 7% when the strain rate changes from 0 to 500 sⳮ1. The O2 mole fraction follows a similar trend. As the strain rate becomes high, the flow residence time becomes short and there is a high O2 leakage to the postflame zone, as a result of the incomplete combustion. However, the change of O2 (and other major species) in the postflame zone is not as strong as the CO concentration. When the strain rate is changed from 0 to 500 sⳮ1, the O2 mole fraction increases by less than 2%. When the flame is close to quenching conditions, the twin flames in the back-to-back configuration move close to each other. Then it is no longer similar to the single-flame turbulent flame condition, and the ‘‘back-to-back’’ flamelet model becomes invalid. In the current afterburner flame, the stretch rate is found to be about 400–600 sⳮ1, far from quenching conditions, therefore, the current flamelet library obtained from the back-to-back configuration should be appropriate. Flamelet Library Approach for Turbulent Premixed Flames The stretched flamelet structure is used to calculate the mean properties of the turbulent flame. Assuming that the joint PDF, 㜷(G, K), is known at the flow field (x, y), the Favre averaged mean quantities at that point can be computed via, for example, for the mass fraction of species i: ⬁
1 ¯ y) ⳮ G, K) Y˜ i(x, y) ⳱ 㜷(G, K)Yi(G(x, q¯ ⳮ⬁ ¯ y) ⳮ G, K)dGdK ⳯ q(G(x,
冮冮
(1)
where the mean density is calculated by q¯ ⳱
⬁
冮冮
ⳮ⬁
¯ 㜷(G, K)q(G(x, y) ⳮ G, K)dGdK
(2)
The joint PDF, 㜷(G, K), is presumed here as the product of a Gaussian distribution for the flame distance function (G) (identical to Ref. [18]) and a lognormal distribution of stretch rate (K), following Ref. [23]. The lognormal distribution is 㜷(K) ⳱
1 ln(K) ⳮ lK exp ⳮ 2 rK 冪2prKK 1
冢 冢
2
冣冣
(3)
Following a similar derivation as in Ref. [23], the mean and variance of the flame strain rate is found to be 2 2 lK ⳱ 1/2(ln(e/k) ⳮ 2rK ) ⳮln(2), rK ⳱ 0.34 (4)
where the mean strain rate is modeled based on the
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TURBULENT COMBUSTION—Measurements in Premixed Turbulent Flames
turbulent kinetic energy and its dissipation rate, that is, e/k. The variance of the flame strain rate, r2K, was obtained experimentally and found to be approximately Reynolds number independent [23]. The current model does not allow negative flame stretch (equation 3). The model can, however, account for partial extinction of the flamelet by including species profiles at the unburned condition in the flamelet library. ¯ (x, y) is the mean distance funcIn equation 1, G ¯ (x, y) ⳱ 0, the tion at flow field point (x, y). If G ¯ (x, y) ⬎ 0 and mean flame surface is then at (x, y). G ¯ (x, y) ⬍ 0 represent the postflame zone and preG heat zone, respectively (Fig. 1). The mean flame position in the turbulent flow field is traced with a level-set formulation, the socalled G-equation [12,16]. Here, the RANS equations and two-equation k-e model are used; the mean G-equation and the modeling of the unknown terms have been described in detail by Peters [16]: ¯ ¯ ¯ G ¯ G G G Ⳮ u˜j ⳱ sT t xj xj xj
冢
1/2
冣
r ⳱ sT/sL Now the wrinkled flamelet library ensemble average is 1 Y˜ i(x, y) ⳱ q¯
(6)
More discussion of turbulent flame speed models and experiments can be found in Ref. [26]. In our previous work [18], several turbulent burning speed models were tested. In order to calculate the PDF of the signed distance, the variance of the distance function, which at the mean flame surface is related to the mean flame thickness, requires further modeling. One may model the variance of G by, for example, a transport equation [16] or using various dimensional arguments [18,27]. The above formulation may be referred to as the ‘‘translating flamelet model.’’ The model was shown [18] to yield reasonably good results for the major species such as CO2, O2, and temperature. It, however, underpredicted the intermediate species, CO. The formulation accounts for fluctuating flamelets (translating motion) in the mean flame zone. It models the effect of flame broadening by turbulence. However, the wrinkling increased flame surface area is not properly modeled. In a recent numerical study of wrinkled random surfaces [27], it was shown that ¯ | K 1. Physically, this in the mean flame brush |ⵜG
⬁
冮冮
ⳮ⬁
¯ G(x, y) ⳮ G ,K r
冢
㜷(G, K)Yi
冣
¯
冢G(x, y)r ⳮ G, K冣dGdK
(5)
Here u˜j is the Favre averaged velocity component in the Cartesian coordinate xj direction. The above equation is only used to compute the mean flame ¯ ⳱ 0. The distance function in the preheat surface, G zone and postflame zone is obtained via numerical ¯ | ⳱ 1, in a so-called solution of the equation |ⵜG reinitialization step [24]. In equation 5, the turbulent flame speed sT has been modeled using an expression similar to that in Ref. [25], sT ⳱ sT Ⳮ 0.46u⬘ Ⳮ 0.2(sLu⬘)1/2
means that due to wrinkling, the inner layer of the flamelet is more densely seen in the mean flame brush. It follows that the mean distance to the flame (i.e., inner layer) is much shorter. This shortened distance to flames is not taken into account by the re-initialized mean G. In the flamelet library model of Peters [16], the wrinkling effect is modeled by dividing a mean distance function by a parameter r ⳱ |ⵜG|G⳱0, the ratio of wrinkled flame surface area to the mean ¯ ⳱ 0. The slope flame surface area conditioned at G ¯ /r in the turbulent flame brush is flatter, since of G r ⬎ 1. As a first approximation, a convenient model for the flame surface area ratio would be
⳯q
(7)
Similarly, the mean density is calculated by q¯ ⳱
⬁
冮冮
ⳮ⬁
¯
冢G(x, y)r ⳮ G, K冣dGdK
㜷(G, K)q
(8) Simulation of CO and Comparison with Experimental Data The experimental flame is shown in Fig. 1. The experiments were conducted by Volvo Aero Corporation aimed at studying the fundamental physics of the premixed combustion in afterburners and providing a database for modeling (thus known as validation rig 1). Fig. 1 shows the experimental setup, which consists of a rectangular, water-cooled channel with height 0.12 m and width 0.24 m. A fuel and air mixture at a temperature of 600 K enters from the left at a mass flow of 0.6 kg/s. The pressure at the combustion chamber is 1 atm. A V-shaped flame is stabilized after the triangular prismatic flame holder. The velocity, temperature, and some species are measured by Laser-Doppler anemometry (LDA) and gas analysis equipment, that is, two non-dispersive IR instruments for CO and CO2, a paramagnetic analyzer for O2, and a flame ionization detector for unburned hydrocarbons. More details of the experiments are found in Ref. [17]. The following numerical calculation is carried out using an in-house computational fluid dynamics (CFD) code [28]. The governing equations are discretized using a hybrid of second-order central and first-order upwind finite difference scheme, and pressure-velocity coupling is through a distributive
CO FORMATION IN PREMIXED TURBULENT COMBUSTION
Fig. 4. Measured and calculated Favre mean mole fraction of O2, CO2, and CO (on a dry basis), as well as temperature along the cross-flow direction at x ⳱ 0.15 m. Exp.: measured data; Num. (1): numerical results using the translating and unstretched flamelet model; Num. (2): numerical results using the translating and stretched flamelet model; Num. (3): numerical results using the wrinkling and stretched flamelet model.
Fig. 5. Measured and calculated Favre mean mole fraction of O2, CO2, and CO (on a dry basis), as well as temperature along the cross-flow direction at x ⳱ 0.350 m. Symbols and lines are identical to Fig. 4.
Gauss-Seidel scheme. As the problem is statistically stationary and two dimensional, a two-dimensional grid is used with a 100 ⳯ 70 grid in the x and y directions, respectively. The computational domain is ⳮ0.1 m ⱕ x ⱕ 0.8 m and ⳮ0.06 m ⱕ y ⱕ 0.06 m. The origin of the x-y coordinate is defined in Fig. 1. The grid resolution was validated by computation on several different grids, and no significant change in solution was found when further refining the grid. The calculated mean flow field based on the k-e model agrees reasonably well with the measured data. The root mean square turbulent velocity fluctuation is about u⬘ ⬃ 10 m/s. The Karlovitz number
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(following the definition of Bradley [26]) ranges from 0.1 to 0.7 in the flame zone, which supports the use of flamelet models. The mean species and temperature distributions are calculated using the translating flamelet model, equation 1, and the wrinkling flamelet model, equation 7, with and without the effect of flame stretch. Figs. 4–5 show the mean mole fraction of O2, CO, and CO2, as well as temperature distribution along the y direction at different positions downstream of the flame holder. In the figures, y ⳱ 0 represents the symmetry plane and y ⳱ Ⳳ0.06 m represents the combustion chamber wall. With the mean flame position and mean flame thickness properly simulated using the level-set G formulation, it is shown that the mean O2 and CO2 as well as temperature profiles at the shown sections can be correctly calculated using both the translating and wrinkling flamelet models. The results are in good agreement with the experimental data. It is interesting to note that the effect of flame wrinkling has a negligible effect on the major species and temperature profiles. From the numerical results, the mean flame stretch in the flow field is found to be about 400–600 sⳮ1. In such moderately low stretched flames, the major species and temperature are not sensitive to the flame stretch (Fig. 3). Figs. 4–5 demonstrate the effect of flame stretch and flame wrinkling on the formation of CO in turbulent flames. As noted, the unstretched flamelet structure yields lower CO mole fractions than the experimental data, both in the postflame zone (near y ⳱ 0 and for large x positions) and in the mean ¯ ⳱ 0, shown as the peak of flame brush (around, G the CO profile). The translating flamelet model with the inclusion of flame stretch does predict the increased non-equilibrium CO concentration in the postflame zone. However, it cannot predict the high CO concentration found in the mean flame brush. With both the flame stretch and the flame wrinkling effect included, as in the wrinkled flamelet model, CO concentrations in the mean flame brush are adequately predicted. Since the definition of G ⳱ 0 in the flamelet coordinate is not unique, it is necessary to evaluate the sensitivity of the results to the definition of G ⳱ 0. Three different definitions have been examined: G ⳱ 0 is defined at (1) the transient point where the net production rate of H radicals changes from negative to positive; (2) at the CH2O peak, and (3) at the CO peak. For a strain rate of 500 sⳮ1, G ⳱ 0 of case (2) is about 0.02 mm upstream of that of case (1), and G ⳱ 0 of case 3 is about 0.05 mm downstream to that of case 1. Let dL denote the difference in the location of G ⳱ 0 from using different definitions of the flame surface. It is shown that dL is in the order of the thickness of a laminar flamelet. Let Y˜i,b denote the mean mass fraction from definition 2. From equation 7, it follows that at point (x, y)
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TURBULENT COMBUSTION—Measurements in Premixed Turbulent Flames
1 Y˜ i ⳱ q¯
⬁
冮冮
ⳮ⬁
¯ ⳮ G G , K r
冢
㜷(G Ⳮ dL, K)Yi
冣
¯ ⳮ G G ⳯q , K dGdK ⬃ Y˜ i,b(1 Ⳮ O(dL/dT)) r (9)
冢
冣
Here dT is the thickness of the turbulent mean flame brush, and it is on the order of the standard deviation of G at point (x, y). The above result is obtained by Taylor expansion of the Gaussian PDF, 㜷(G Ⳮ dL, K), in G-coordinate. Since dL K dT, it is concluded that different definitions of G ⳱ 0 have little influence on the calculated mean species and temperature. Conclusions From the above discussion, the following can be concluded. 1. Flame stretch (aerodynamic strain rate) has a fairly large influence on the laminar burning velocity. It affects the flame structure and the formation of intermediates, such as CO, in lean premixed propane/air combustion, particularly when the stretch rate is high. For moderately low flame strain rate as considered here, the influence of stretch is not significant on the major species and temperature distributions. 2. Flame wrinkling is found to significantly affect the CO profiles in the mean flame brush. Without the inclusion of the effect of flame wrinkling, the high CO concentrations found in the experiments cannot be explained. The effect of flame stretch on CO is mainly in the postflame zone. 3. The major species and temperature levels are found to be not sensitive to flame wrinkling. Both the translating and wrinkling flamelet library models give reasonably good results as compared with experimental data. Acknowledgment This work is supported by the Swedish National Energy Administration’s Consortium for Gas Turbine Center (GTC) and the Swedish Research Council for Natural and Engineering Science (VR).
REFERENCES 1. Correa, S. M., Proc. Combust. Inst. 27:1793–1807 (1998). 2. Peters, N., and Williams, F. A., Combust. Flame 68:185–207 (1987).
3. Seshadri, K., and Peters, N., Combust. Flame 81:96– 118 (1990). 4. Clavin, P., and Williams, F. A., J. Fluid Mech. 116:251– 282 (1982). 5. Pelce, P., and Clavin, P., J. Fluid Mech. 124:219–237 (1982). 6. Matalon, M., and Matkowsky, B. J., J. Fluid Mech. 124:239–259 (1982). 7. Law, C. K., Proc. Combust. Inst. 22:1381–1402 (1988). 8. Rogg, B., Combust. Flame 73:45–65 (1988). 9. Rogg, B., and Peters, N., Combust. Flame 79:402–420 (1990). 10. Candel, S. M., and Poinsot, T. J., Combust. Sci. Technol. 70:1–15 (1990). 11. Chen, J. H., and Im, H. G., Proc. Combust. Inst. 27:819–826 (1998). 12. Williams, F. A., ‘‘Turbulent Combustion,’’ in The Mathematics of Combustion (J. Buckmaster, ed.), SIAM, Philadelphia, 1985, p. 97. 13. Bray, K. N. C., Libby, P. A., and Moss, J. B., Combust. Flame 61:87–102 (1985). 14. Candel, S., Veynante, D., Lacas, F., Darabiha, N., and Rolon, C., Combust. Sci. Technol. 98:245–264 (1994). 15. Hawkes, E. R., and Cant, R. S., Combust. Flame 126:1617–1629 (2001). 16. Peters, N., Turbulent Combustion, Cambridge University Press, Cambridge, UK, 2000. 17. Sjunnesson, A., Olovsson, S., and Sjoblom, B., Validation Rig—A Tool for Flame Studies, Volvo Flygmotor Internal Report VFA 9370-308, Volvo, Trollha¨ttan, Sweden, 1991. 18. Nilsson, P., and Bai, X. S., Exp. Therm. Fluid Sci. 21:87–98 (2000). 19. Peters, N., ‘‘Flame Calculations with Reduced Mechanisms—An Outline,’’ in Lecture Notes in Physics m15 (N. Peters and B. Rogg, eds.), Springer-Verlag, Berlin, 1993, pp. 3–14. 20. de Goey, L. P. H., and ten Thije Boonkkamp, J. H. M., Combust. Flame 119:253–271 (1999). 21. Sun, C. J., Sung, C. J., He, L., and Law, C. K., Combust. Flame 118:108–128 (1999). 22. Mauss, F., ‘‘Entwicklung eines kinetischen Modells der Russbildung mit schneller Polymerisation,’’ Ph.D. thesis, RWTH Aachen, Aachen, Germany, 1998. 23. Abdel-Gayed, R. G., Bradley, D., and Lau, A. K. C., Proc. Combust. Inst. 22:731–738 (1988). 24. Sussman, M., Smereka, P., and Osher, S., J. Comput. Phys. 114:146–159 (1994). 25. Muller, C. M., Breitbach, H., and Peters, N., Proc. Combust. Inst. 25:1099–1106 (1994). 26. Bradley, D., Proc. Combust. Inst. 24:247–262 (1992). 27. Nilsson, P., ‘‘Level-Set Flamelet Library Model for Premixed Turbulent Combustion,’’ Ph.D. thesis, Lund Institute of Technology, Lund, Sweden, 2001. 28. Bai, X. S., and Fuchs, L., Comput. Fluids 23:507–521 (1994).
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COMMENTS Fenshan Liu, National Research Council, Canada. Have you compared your predictions of other intermediate species, such as hydrogen, with experimental data? Author’s Reply. From our numerical calculations, in addition to carbon monoxide, other intermediate species, such as hydrogen, were found to be very sensitive to the flame wrinkling as well. However, in the current flame, hydrogen and other intermediate species were not measured; therefore, no comparison of these species with the experimental data was conducted. ● Andrei Lipatnikov, Chalmers University of Technology, Sweden. My understanding is that upon numerically solving the G-equation, the solution is corrected by thickening
the flame artificially. Does this mean that the G-equation as used cannot predict the experimental data in this case? Author’s Reply. The level-set G-equation was derived and should be used to calculate the location of the flame surface only. If the G-equation is used in DNS, the fluctuation and wrinkling of the flame surface should be captured correctly, provided the stretched local laminar flame burning velocity is properly calculated/modeled. The time averaged, mean flame thickness is then calculated directly from the G-equation DNS data. In the current RANS framework, the mean G-equation is used to calculate the mean flame location, not the mean flame thickness. The mean flame thickness, which is related to the fluctuation of the instantaneous flame surface, can be obtained from the variance of G. Some models for the variance of G can be found in (Refs. [16,18] in paper).