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Measured burning velocities of stretched inwardly propagating premixed flames
Proceedings of the Combustion Institute, Volume 28, 2000/pp. 1783–1791
MEASURED BURNING VELOCITIES OF STRETCHED INWARDLY PROPAGATING PREMIXED FLAMES ALFONSO F. IBARRETA and JAMES F. DRISCOLL Department of Aerospace Engineering University of Michigan Ann Arbor, MI 48109, USA
The propagation of an inwardly propagating laminar propane/air flame was studied by forming a pocket of reactants that is surrounded by a premixed flame. A flame-vortex interaction initially creates the pocket, which persists long after the vortex is gone. Very large negative stretch rates were achieved such that the Karlovitz number exceeded unity. The local burning velocity was determined from high-speed shadowgraph images and was correlated with the local stretch rate, which is entirely due to curvature. Results were compared to the predictions of asymptotic analysis and to experimental results from outwardly propagating flames. The Markstein number that was determined from the data was found to be more than twice the value previously reported for positively stretched flames. Burning velocities were as large as seven times the unstretched value. These results are consistent with DNS simulations and Bunsen tip experiments, where negative curvature has been shown to have a larger effect than predicted by asymptotic analysis.
Introduction The understanding of the structure and behavior of laminar flames that undergo flame stretch is crucial to the proper modeling of turbulent flames [1]. Experimental and numerical studies have shown that the regions of negative curvature in turbulent flames are just as probable as those of positive curvature, since curvature probability density functions are nearly symmetric around zero [2,3]. The regime of negative stretch has been extensively studied using numerical methods by Sun et al. [4,5], Chen et al. [6], Haworth and Poisont [7], and Bradley et al. [8]. Unfortunately, negative flame stretch is difficult to replicate in the laboratory. Most of the experimental data on negatively stretched flames was obtained from velocity measurements made near the tip of Bunsen burner flames by Echeckki and Mungal [9] and others [10,11]. However, the flowfield inside the Bunsen flame tip is complex, since both strain and curvature effects are present, and the spatial resolution requirements of such measurements are severe. The present study focuses on an inwardly propagating flame (IPF). For such a flame, the stretch rate is negative and is entirely due to curvature; strain effects and velocity gradients in the reactants are negligible. Pockets of reactants surrounded by a flame are naturally caused in turbulent flames by the interaction of vortical structures with the flame surface. Therefore, it was decided to form an IPF by this natural method; a vortex is passed through a laminar flame and an inwardly burning pocket of reactants is formed. At a much later time, when the
vortex has long gone, the final stages of the pocket burnup process are studied. The main objective of the present study is to measure the correlation between the local burning velocity and the local flame stretch, which is entirely due to negative flame curvature, for a laminar IPF. The observed correlation for IPF cases will be compared to experimental results for outwardly propagating flames (OPFs). Effects of Flame Stretch The relationship between the the laminar burning velocity and the flame stretch was proposed by Markstein [12] to be Su ⳱ Suo ⳮ LK
(1)
Su is the local burning velocity, defined as the propagation velocity of the stretched flame with respect to the unburned reactants. Sou is the unstretched laminar burning velocity, and L is the Markstein length. The stretch rate K is defined as 1/A(dA/dt) [13] and describes the growth of an infinitesimal flame surface. The Markstein length is proportional to a characteristic flame thickness based on the mass diffusivity of the deficient reactant, dD, so that a dimensionless Markstein number can be defined as Ma ⳱ L/dD [14]. The stretch rate K can be non-dimensionalized as well by the characteristic flow time within the flame (residence time) to obtain the Karlovitz number Ka. This characteristic time has been defined by either using the unstretched [13] or stretched [14,15] values of the burning velocity and flame thickness. The
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Equation 4 has been shown [13] to be valid at low stretch rates and assumes a simple linear relation between the stretch and the burning velocity. Equation 5 is a nonlinear relationship between laminar burning velocity and the Karlovitz number introduced by Kwon et al. [15] to extend the applicability of equation 1 to finite levels of stretch [16]. Equation 5 has been shown to agree with experimental data for outwardly propagating spherical flames at low to intermediate stretch rates, for Ka* up to 0.7 [14,17]. One goal of this study is to determine if relations 4 or 5 agree with experimental data for IPFs. It is noted that both equations 4 and 5 give the same Markstein number at the limit of small stretch (K 0), when the stretched burning velocity approaches its unstretched value.
Stretch Rate of IPFs Fig. 1. Two theoretical predictions of the effects of flame curvature on the burning velocity (Su). Plot of equation 7 (—), which correlates the data of Kwon et al. [15] and equation 8 (– –), which is derived from asymptotic analysis of Law [13]. Both are calculated for a Markstein number of 7.0 which is the observed Markstein number for OPF with U ⳱ 0.65. Several contours of Ka* are included to give an idea to the amount of stretch imposed on the flame.
flame thickness (dD) has been defined as the ratio of mass diffusivity (Du) to the stretched burning velocity (Du/Su) [15] or as dD, which is the ratio Du/Sou [13]. In the case of combustion in air, the mixtures are highly diluted, so that Du is approximated as the binary diffusivity of the fuel with respect to the diluent (nitrogen), which equals 0.113 cm2/s for the mixture used [14]. Law [13] defines the Karlovitz number (Ka*) using the unstretched laminar burning velocity, as follows: Ka* ⳱
冤(SD ) 冥 K u o 2 u
(2)
Kwon et al. [15] used the stretched value of the burning velocity to define the following Karlovitz number:
冤 冥
Du Ka ⳱ K (Su)2
(3)
Two different relationships for laminar burning velocity that have been widely used to obtain Markstein numbers from experimental studies are: Su ⳱ 1 ⳮ MaKa* Sou
(4)
Su 1 ⳱ Sou 1 Ⳮ Ma Ka
(5)
For an ideal inwardly propagating spherical flame, the velocity of reactants inside the pocket is negligibly small because of symmetry requirements. The hydrodynamic strain rate is • u¯ ⳮ n¯ • (n¯ • )u¯ in the reactants just upstream of the flame; therefore, a spherical IPF is not subjected to hydrodynamic strain. Based on relations reported by Law [13], the stretch rate for such an IPF is K⳱
Su Rc
(6)
where Su is the local burning velocity and Rc is the radius of curvature of the flame surface. The radius of curvature is defined to be negative if the surface is concave toward the reactants, which occurs for IPF cases. Equation 6 can be combined with equations 2 through 5 to obtain two possible relations between the stretched burning velocity and the local flame curvature [13,15] Su 1 ⳱ Sou 1 Ⳮ Ma d oD/Rc
(7)
Su d oD o ⳱ 1 ⳮ Ma Su Rc
(8)
where d oD is the unstretched laminar flame thickness. Fig. 1 shows the differences between equations 7 and 8 for an assumed Markstein number of 7.0, which is the Markstein number extrapolated from OPF measurements for the current propane/air mixture (U ⳱ 0.65). The figure shows that equation 7 predicts a nonlinear increase of the local burning velocity as the pocket burns inward (for positive Ma numbers). However, equation 8 predicts that the velocity will increase linearly until the pocket is consumed. The present data are used to determine which of the two curves in Fig. 1 best represents the experimental observations.
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Fig. 2. Schematic of the combustion chamber showing pocket formation (left diagram) and the positions of the vortex and flame at an earlier time prior to the interaction (right diagram).
Experimental Setup The experiment that was used is shown in Fig. 2; a vortex-flame interaction creates a pocket of reactants in a premixed propane/air mixture at equivalence ratios (U) ranging from 0.58 to 0.65. Roberts and Driscoll [16] and Roberts et al. [17] report numerous images of pockets that were formed with this device. An upward moving laminar flame interacts with a downward moving toroidal vortex, causing the flame to wrap around the vortex. The distorted flame surface will eventually pinch off, so that a small bubble of unburned reactants will be completely surrounded by the flame surface. This pocket is studied long after the vortex has dissipated. The current setup consists of two rectangular chambers, which are joined together but initially separated by a removable splitter plate (Fig. 2). The lower chamber is where the interaction takes place. It contains an ignition source, exhaust ports, and quartz windows for optical access. The upper chamber is joined to a loudspeaker and contains a plate with a 4 cm diameter orifice (where the toroidal vortex is formed) and a pair of solenoid-operated pressure relief ports. By using two chambers, it is possible to separately control the equivalence ratio inside the vortex and in a region far outside the vortex. The mixture inside the vortex is kept very lean, while the mixture far from the vortex is a slightly richer propane/air mixture that will assist the flame to rapidly envelop the pocket formed by the vortex.
In this fashion, larger, slower burning pockets can be achieved without the use of strong vortices, which might not dissipate completely, which is a requirement of this experiment. The chambers are originally separated by a splitter plate, which is removed before the experiment is run. First, the exhaust port is opened, to prevent pressure buildup inside the chamber. A toroidal vortex is produced by driving the speaker with a rampvoltage generator. The vortex is formed in the upper chamber and will thus entrain only fluid contained in this section. The experiment is carefully timed so that the initial vortex/flame interaction occurs just after the vortex crosses the dividing line between both chambers. This measure reduces the amount of outside fluid entrained into the vortex. The difference in U between both chambers is also kept within 15% so that any small amount of mixing will have little effect on the equivalence ratio of the pocket. Using the vortex theory of Maxworthy [18], it was estimated that less than 20% of the total pocket volume is entrained from the outside gas mixture; therefore, the uncertainty in U is 3%. A spark creates a flame which travels upward through the bottom chamber. The pocket is initially tear shaped and has an initial diameter of 1 cm, and it becomes nearly spherical as its diameter decreases. Data Reduction Flame speeds were measured by recording shadowgraph images of the pocket burnup at 1500
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frames per second using a Fastex 16 mm high-speed movie camera. Images are digitized using a Kodak CCD camera with 2000 ⳯ 3000 pixel array. The digitized flame surface is fitted with a high-order polynomial. Fig. 3 shows several snapshots of the flame surface at different times. The local burning velocity (Su) was determined from Su ⳱ Dxn/Dt
Fig. 3. Flame surface at three times, showing the normal to the surface at two locations. U ⳱ 0.58.
Fig. 4. Measured burning velocities of IPFs for U ⳱ 0.62, Sou ⳱ 16.0 cm/s, and doD ⳱ 0.071 mm. Solid symbols represent results obtained using equation 9; open symbols were obtained using equation 11. The plot includes the predictions of equation 8 (– –) and equation 7 (– • –) utilizing a Markstein number of 7.3 that is interpolated from the data of Tseng et al. [14] for the present value of U. Several contours of Ka* were added to indicate the amount of stretch imposed on the flame.
(9)
where Dxn is the distance from one flame front to the next (Fig. 3) along the line normal to the initial surface, and Dt is the time between images [19]. The gas velocity inside the pocket is negligibly small, as discussed below. The local flame curvature and burning velocity are calculated at 400 points along the surface for each image. The curvature is defined as the average of the curvatures at the initial and final positions during flame travel. The three-dimensional radius of curvature Rc, is defined by [20] 1 1 1 ⳱ Ⳮ Rc R1 R2
(10)
where R1 is the component of curvature in a plane cutting through the center of the pocket (defined to be the inverse of the curvature of the two-dimensional flame surface). R2 is the out-of-plane component, which is the distance of the flame surface element to the axis of symmetry, Rsymm divided by cos h, where h is the angle between the normal vector and the axis of symmetry. Note that for a spherical flame both R1 and R2 are equal to the measured radius of the sphere (Rf), so Rc ⳱ Rf/2. It is important to ensure that at the time when measurements of the burning velocity are made, the vortex has attenuated such that the gas velocity inside the pocket is negligible with respect to the flame speed. The initial vortex will rapidly vanish due to the high viscosity caused by the flame and the dilation of the gas [3,21,22]. The velocity inside a typical pocket was measured using particle-imaging velocimetry (PIV) in a previous study [23]. A characteristic induced velocity Uchar was determined by dividing the total circulation inside the pocket (C) by 2pd, where d is the vortex diameter. From the PIV measurements, it was determined that at the time of pocket formation, only about 10% of the induced velocity remains and quickly dissipates before pocket burnout. The vortices used in the present work were very weak and had a characteristic initial vortex velocity of Uh 13 cm/s, which will decay to less than 1.3 cm/s at the moment of pocket formation. Since the burning velocities measured were 6.5 cm/s to 80 cm/s, the uncertainty in flame speed measurement that arises from a non-zero reactant velocity varies from 1.6% to 20%.
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Results
Fig. 5. Measured burning velocities of IPFs for U ⳱ 0.65, Sou ⳱ 18.6 cm/s and doD ⳱ 0.061 mm. Solid symbols represent results obtained using equation 9; open symbols were obtained using equation 11. The plot includes the predictions of equation 8 (– –) and equation 7 (– • –) utilizing a Markstein number of 7.0 that is interpolated from the data of Tseng et al. [14] for the present value of U. Several contours of Ka* were added to indicate the amount of stretch imposed on the flame.
Figure 4 shows results for the U ⳱ 0.62 flame; the non-dimensional burning velocity is plotted against the non-dimensional flame curvature for the last eight frames of the interaction. If the relation used by Kwon et al. (equation 8) to correlate their OPF data applies to the IPF, the data points in Fig. 4 would lie on a line that passes through Su/Sou ⳱ 1 and would have a slope equal to Ma. In Fig. 4, about 2000 points are plotted using a moving-average scatter plot. The scatter, shown by the standard deviation brackets, is significant due to measurement uncertainties discussed later, but basic trends can be deduced. The lower dotted line in Fig. 4 represents equation 8 for a Markstein number of 7.3, which is the value extrapolated from data obtained Tseng et al. [14] for an OPF. The upper dotted curve in Fig. 4 is equation 7 for the same Markstein number. The IPF data points do not match either of these relations but instead follow a straight line (in agreement with equation 8) with a slope of 22.8, which is the Markstein number for the IPF at U ⳱ 0.62. It is important to note that Fig. 4 shows that there is a large increase of the stretched burning velocity in the present experiment, which is as large as six times the unstretched value. A second method to determine the burning velocities was used, which involved the volume, V, and surface area, A, of the flame during the final phase of pocket burnout. During the last few frames, the flame pocket is nearly spherical, so that a global value of Su and Rc can be obtained using the following equations (valid only for a sphere):
冢
Fig. 6. Unstretched laminar burning velocity of propane/air flames obtained by Law et al. [24] and Vagelopolous et al. [25]. The observed extinction conditions in the present experiment (䉴) are included as reference.
1/3
冣
dV 1 1 3 Su ⳱ ⳮ and Rc ⳱ ⳮ V dt A 2 4p
(11)
The volume, V, was determined from a numerical integration using the surface boundary (Fig. 3) and assuming axisymmetry. Results for this second method are shown as triangles in Fig. 4, and produce final Markstein numbers that agree with values obtained using equation 9 to within 10%. Fig. 5 is an identical plot showing the results for U ⳱ 0.65; the experimental Ma of 19.3 is considerably larger than the value of 7.0 determined from OPF experiments. The data in Figs. 4 and 5 show that the present results are in good agreement with the linear relation given by equation 8; however, the Markstein number for an IPF is measured to be two to three times larger than that of a corresponding OPF. Fig. 6 is a plot of the unstretched laminar burning velocity, Sou versus equivalence ratio that was obtained from data by Law [24] and Vagelopolous et al. [25]. The values of Sou needed to normalize the measurements for equivalence ratio of U ⳱ 0.58 were obtained by extrapolating the data from Fig. 6.
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Fig. 7. Comparison of experimentally obtained Markstein numbers for present IPFs with Markstein numbers obtained from several OPF experiments and numerical simulations of Tseng et al. [14], Taylor et al. [26], and Sun et al. [4]. Each open circle represents the average of 2000 flame segments obtained from one run (such as in Fig. 5); solid circles are the average of multiple runs.
Comparisons to Other Negatively Stretched Flames Figure 7 shows how the values of Markstein number deduced for the IPFs (using equation 8) compare with values reported for the OPFs studied by Tseng et al. [14], Taylor et al. [26], and Sun et al. [4]. It is seen that the present values of Ma for IPF cases are two to three times larger than those reported for the OPFs. While the disagreement between the IPF and OPF cases in Fig. 7 cannot yet be adequately explained, there have been several other studies which have reported larger values of Ma for an negatively stretched flame than the reported values for a corresponding OPF. Echekki and Mungal [9] measured velocities near the tip of a Bunsen flame and reported that the slope of their measured curve of Su versus flame curvature is 6.46 for a stoichiometric methane/air mixture, which yields a value of Ma of 6.46, based on equation 7. Tseng et al. [27] report that the Markstein number for their OPF in this same mixture is 1.3. Thus, the Markstein number of Echekki and Mungal’s IPF is five times the OPF value reported by Tseng et al. for the same fuel/air mixture. In the numerical study of Echekki and Chen [3], the propagation speeds of curved flames were computed using a reduced four-step mechanism. They reported a curve similar to Fig. 4 in which the propagation speed was plotted against the flame curvature; they found that the slope of this
curve was more than three times larger in the negative curvature region than in the positive curvature region. In a related computational study of curved flames, Chen and Echekki [28] showed that near a cusp (where the curvature is negative), their computed local propagation speeds were more than 50 times the unstretched value (which will tend to result in large Markstein numbers). Negative curvature was shown by Chen and Echekki to focus certain light radicals, such as H, O, and OH, that are important in the chain-carrying reactions, which will result in increased local heat release and larger burning velocities. Numerical studies by Sun et al. [4] showed good agreement between OPFs and IPFs at very low stretch rates. It is not yet clear whether the differences between their Markstein numbers and the present values are due to finite stretch rates, preheating of gas inside the pocket, or complex chemistry.
Experimental Uncertainties and Effects of Flame Thickness A large amount of experimental uncertainty is observed in Figs. 4 and 5, as noted by the large error bars, which denote the root-mean-squared deviations of the data when 2000 flame segments are considered. Most of this uncertainty is due to properly aligning the individual images to obtain the overlaid image such as the one shown in Fig. 3. The pocket of reactants moves downward, carried by the bulk of the exhaust, as the flame burns inwardly. To remove the effects of this convection, the images had to be moved along the axis of symmetry to obtain Fig. 3. It is noted that there is no alignment problem when equation 11 was used to determine the triangle symbols in Figs. 4 and 5, since equation 11 only requires information about the volume and surface area of the pocket and not its location relative to other images. However, equation 11 could only be used in the late stages of pocket burnup when the pocket is nearly spherical. Additional uncertainties are introduced because the flame surface is fitted with a polynomial and because the flame boundary has a finite thickness on the digitized shadowgraph images. The flame boundary is defined as the line between the white line and the black line that appears on the shadowgraph; this line is the location of the maximum value of the first derivative of the gas density. The initial image has a viewing area of 3.6 ⳯ 2.4 cm and is digitized by a 3000 ⳯ 2000 pixel array, which provided a spatial digitization uncertainty of 3.6 cm/ 3000, which is 12 lm and is too small to be a source of error. The uncertainty associated with identifying the flame boundary, due to shadowgraph limitations
STRETCHED INWARDLY—PROPAGATING PREMIXED FLAMES
and curve fitting the surface, is approximately 4 pixels or about 48 lm. The typical displacement of the flame segment is 0.437 mm in Fig. 3. So the ratio of spatial uncertainty to displacement is about 10% and is not a significant contributor to the much larger uncertainties shown in Fig. 4. The ratio of the flame thickness to the pocket diameter (dD/df) for the current experiment ranges from 0.002 to 0.025. Thus, the flame thickness (Du/ Su) is always less than 3% of the pocket diameter. The flame thickness decreases by a factor of ten as the pocket becomes smaller for the present conditions, because the flame speed increases by a factor of ten, as was shown in Fig. 4. Not all flames behave this way; for fuel/air combinations that have a negative Markstein number, the flames would not become thinner as the pocket becomes smaller. If the ratio (dD/df) approaches unity (which it does not in the present experiment), then the curvature of the leading edge of the flame differs from that of the trailing edge, and the unsteady accumulation of mass inside the flame must be considered when writing the equation for the conservation of mass, as discussed by Sun et al. [4].
Conclusions 1. The present experiment is shown to provide a useful way to create a repeatable and controllable IPF which is tear shaped initially and becomes nearly spherical in the later stages. IPFs provide a way to measure the effects of negative flame curvature accurately and without the added complexity of hydrodynamic strain. 2. The burning velocity–curvature correlation follows a linear trend, in agreement with equation 8 adequately explains the results; however, the slope of the correlation curve yields a Markstein number that is two to three times larger than the Markstein number reported previously for OPFs in the same fuel/air mixture. 3. The reasons why the Markstein number is different for the IPFs and OPFs is not known, but the present findings are consistent with previous studies of Bunsen tips and with numerical simulations which have reported similar trends. Acknowledgments Support for this research was provided by National Science Foundation grants CTS 9123834 and CTS 9904198, administered by Dr. Farley Fisher, and NASA grant NCC3656.
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REFERENCES 1. Peters, N., Proc. Combust. Inst. 21:102:1231–1250 (1986). 2. Lee, T.-W., North, G. L., and Santavica, D. A., Combust. Flame 93:445–456 (1993). 3. Echekki, T., and Chen, J. H., Combust. Flame 106:184–202 (1996). 4. Sun, C. J., Sung, C. J., He, L., and Law, C. K., Combust. Flame 118:108–128 (1999). 5. Sun, C. J., and Law, C. K., Proc. Combust. Inst. 27:963–970 (1998). 6. Chen, J. H., and Im, H. G., Proc. Combust. Inst. 27:711–717 (1998). 7. Haworth, D., and Poinsot, T., J. Fluid Mech. 244:405– 436 (1992). 8. Bradley, D., Gaskell, P., and Gu, X., Combust. Flame 104:176–198 (1996). 9. Echekki, T., and Mungal, M. G., Proc. Combust. Inst. 23:455–461 (1990). 10. Poinsot, T., Echekki, T., and Mungal, M. G., Combust. Sci. Technol. 81:45–73 (1992). 11. Wagner, T. C., Combust. Flame 59:267–272 (1985). 12. Markstein, G. H., J. Aeronaut. Sci. 18:199–209 (1951). 13. Law, C. K., Proc. Combust. Inst. 22:1381–1402 (1988). 14. Tseng, L.-K., Ismail, M. A., and Faeth, G. M., Combust. Flame 95:410–426 (1993). 15. Kwon, S., Tseng, I.-K., and Faeth, G. M., Combust. Flame 90:230–246 (1992). 16. Roberts, W. L., and Driscoll, J. F., Combust. Flame 87:245–256 (1991). 17. Roberts, W. L., Driscoll, J. F., Drake, M. C., and Goss, L. P., Combust. Flame 94:58–69 (1993). 18. Maxworthy, T., J. Fluid Mech. 51:15–32 (1971). 19. Sinibaldi, J. O., Mueller, C. J., and Driscoll, J. F., Proc. Combust. Inst. 27:827–832 (1998). 20. Matalon, M., Combust. Sci. Technol. 31:169–181 (1991). 21. Poinsot, T., Veynante, D., and Candel, S., J. Fluid Mech. 228:561–606 (1991). 22. Mueller, C. J., “Measurements of a Flame-Vortex Interaction Dynamics and Chemistry,” Ph.D. thesis, University of Michigan (1996). 23. Mueller, C. J., and Driscoll, J. F., Proc. Combust. Inst. 26:347–355 (1996). 24. Law, C. K., Zhu, D. L., and Yu, G., Proc. Combust. Inst. 21:1419–1426 (1998). 25. Vagelopolous, C. M., Egolfopoulus, F. N., and Law, C. K., Proc. Combust. Inst. 25:1341–1345 (1994). 26. Taylor, S. C., Ph.D. thesis, University of Leeds, U.K., 1991. 27. Aung, K., Tseng, L.-K., Ismail, M., and Faeth, G. M., Combust. Flame 102:526–530 (1995). 28. Chen, J. H., and Echekki, T., Combust. Flame 116:15– 48 (1999).
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COMMENTS Stephen Klotz, Princeton University, USA. You described how you account for internal circulation (caused by the vortex generation process). Have you accounted for the relative motion of your flame product and the surrounding gas in your chamber? If so, how does this flow affect your flame speed measurements? Have you considered using PIV (or a similar technique) to determine the important fluid motions in your flame generation device?
decreased to 70 mm. So the range of the present measurements, the ratio of the thermal thickness to the flame diameter varies from 0.09 to 0.21. These numbers indicate that little, if any, overlap of preheat zones occur when the measurements were made.
Author’s Reply. The flame propagation speed is defined as the speed of the combustion wave with respect to the reactants. Therefore, the relative motion of the flame pocket and the surrounding gas will not have any affect on the propagation speed, unless the surrounding gas somehow causes a relative motion between the reactants inside the pocket and the thin flame zone. As the pocket moves out of the chamber, both the pocket of reactants and the surrounding flame are convected at the mean flow velocity. Any velocity field that is uniformly superimposed on both the thin flame zone and the reactants will, by definition, have no effect on the propagation speed. There is no physical reason (and we have no evidence) to suggest that the vertical convection speed of the flame should differ from that of the pocket of reactants. The fact that our measured propagating speeds approach the known value of the unstretched propagation speed for small values of stretch indicates that the measurement procedure is sound.
1. Bechtel, J. H., Blint, R. J., Dasch, C. J., and Weinberger, D. A., Combust. Flame 42:197–213 (1981).
● Jacqueline H. Chen, Sandia National Laboratories, USA. As the flame burns inward, at some point the enhancement of the flame propagation is influenced by mutual annihilation. Can you quantify the importance of this effect and at what radius of curvature it becomes significant? Author’s Reply. Yes, as two segments on opposite sides of the spherical flame approach each other, their preheat zones eventually will overlap, which can greatly increase the propagation speeds of each segment. However, the present measurements were made before this overlap begins. We note that this overlap is fundamental property that possibly can occur in all flames with negative curvature, including inwardly propagating flames, bunsen-tip flames, and certain segments of turbulent flames. It may explain why the present results differ from those reported for outwardly propagating flames. The thermal thickness of the preheat zone is Tmax ⳮ Tmin)/(T/x)max which is measured to be 7.4 ␣/SL in laminar flames [1]. Based on this estimate, the thermal thickness of the flame described in Fig. 5 is 0.30 mm when the flame diameter is 3.40 mm (when the earliest measurements are made). At a later time when the final measurements are made, the estimated thermal thickness has decreased to 0.15 mm and the flame diameter has
REFERENCE
● Yiguang Ju, Princeton University, USA. Since laminar flame speed depends on the local velocity normal to the flame front on the unburned side and on the local flame structure, how do you make sure that the flow in the unburned pocket is zero and the structure of the fine evoluting local flame structure is the same as the ideal inward propagating flame? Author’s Reply. Steps were taken to ensure that the vortex inside the pocket has decayed sufficiently such that any gas velocity in the reactants is less than 10% of the flame propagation speed and therefore can be neglected in determining the propagation speed. First, a sufficiently weak vortex was chosen such that it will decay to negligible strength before the pocket is formed. Secondly, the time for the vortex to decay completely. Thirdly, PIV measurements from a previous study [1] were used to quantify the velocity of reactants inside the pocket. Based on the previous PIV measurements of [1], the maximum possible error introduced by reactant motion is 1.4 cm/s. The measured propagation speeds in Fig. 5 vary from 32 cm/s to 64 cm/s, so this error introduced by reactant motion varies from 2% to 4%.
REFERENCE 1. Mueller, C. J., Driscoll, J. F., Reuss, D. L., Drake, M. C., and Rosalik, M. E., Combust. Flame 112:342–358 (1998). ● William Roberts, North Carolina State University, USA. How do you ensure the apparatus is isobaric? If it is not, then you would have compression of the flame pocket, leading to a higher apparent SL. Could this explain your higher Ma number? One of your possible explanations for higher Ma number is preheating of the pockets. If this were true, the pocket would expand, leading to a lower apparent
STRETCHED INWARDLY—PROPAGATING PREMIXED FLAMES SL. Can you give a weighting of SL increase due to increasing To and SL decrease due to expansion? Author’s Reply. 1. The pressure inside the chamber does not increase by more than approximately 10% of its initial value of 1 atm. There are large exhaust ports which remain open at one end of the chamber, and there are pressure relief valves which are open at the other end of the chamber. The very lean equivalence ratios used (0.55 to 0.6) and the slow movement of the flame (10 cm/s) also prevent significant pressure rise. 2. The present flames become thinner as they burn inward, which would tend to prevent merging of the preheat zones and reduce any preheating of the reactants. However, it is not known if preheating of the reactants does occur. If the temperature of the reactants is increased by 60 C, for example, this would approximately double the unstretched burning velocity but would only increase the radius of the pocket by 6%. Therefore, the effects of gas density changes are negligible compared to the sensitivity of the flame speed to the temperature of the reactants. We note that if the reactants are being
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heated, this would be a fundamental characteristic of inwardly propagating flames, and may explain why such flames may have different properties than outwardlypropagating flames. Such heating, if it occurs, would not affect the accuracy of the present measurements or the generality of the results for IPFs. ● Arvind Atreya, University of Michigan, USA. This is excellent experimental work. I wonder if you have studied fuels other than propane to see the Lewis number effect? Author’s Reply. No, we have not. It is difficult to achieve a nearly spherical flame speed as the pocket of fuel burns out. Lean propane-air flames become spherical in this experiment because they are diffusionally stable and do not have small scale wrinkles. Increasing the equivalence ratio creates less stable flames which have small wrinkles, which cause a non-spherical pocket. Future work is planned to study lean methane-air flames, but since they are diffusionally unstable and wrinkle easily, it may be difficult to achieve a final flame shape that is spherical.
MEASURED BURNING VELOCITIES OF STRETCHED INWARDLY PROPAGATING PREMIXED FLAMES ALFONSO F. IBARRETA and JAMES F. DRISCOLL Department of Aerospace Engineering University of Michigan Ann Arbor, MI 48109, USA
The propagation of an inwardly propagating laminar propane/air flame was studied by forming a pocket of reactants that is surrounded by a premixed flame. A flame-vortex interaction initially creates the pocket, which persists long after the vortex is gone. Very large negative stretch rates were achieved such that the Karlovitz number exceeded unity. The local burning velocity was determined from high-speed shadowgraph images and was correlated with the local stretch rate, which is entirely due to curvature. Results were compared to the predictions of asymptotic analysis and to experimental results from outwardly propagating flames. The Markstein number that was determined from the data was found to be more than twice the value previously reported for positively stretched flames. Burning velocities were as large as seven times the unstretched value. These results are consistent with DNS simulations and Bunsen tip experiments, where negative curvature has been shown to have a larger effect than predicted by asymptotic analysis.
Introduction The understanding of the structure and behavior of laminar flames that undergo flame stretch is crucial to the proper modeling of turbulent flames [1]. Experimental and numerical studies have shown that the regions of negative curvature in turbulent flames are just as probable as those of positive curvature, since curvature probability density functions are nearly symmetric around zero [2,3]. The regime of negative stretch has been extensively studied using numerical methods by Sun et al. [4,5], Chen et al. [6], Haworth and Poisont [7], and Bradley et al. [8]. Unfortunately, negative flame stretch is difficult to replicate in the laboratory. Most of the experimental data on negatively stretched flames was obtained from velocity measurements made near the tip of Bunsen burner flames by Echeckki and Mungal [9] and others [10,11]. However, the flowfield inside the Bunsen flame tip is complex, since both strain and curvature effects are present, and the spatial resolution requirements of such measurements are severe. The present study focuses on an inwardly propagating flame (IPF). For such a flame, the stretch rate is negative and is entirely due to curvature; strain effects and velocity gradients in the reactants are negligible. Pockets of reactants surrounded by a flame are naturally caused in turbulent flames by the interaction of vortical structures with the flame surface. Therefore, it was decided to form an IPF by this natural method; a vortex is passed through a laminar flame and an inwardly burning pocket of reactants is formed. At a much later time, when the
vortex has long gone, the final stages of the pocket burnup process are studied. The main objective of the present study is to measure the correlation between the local burning velocity and the local flame stretch, which is entirely due to negative flame curvature, for a laminar IPF. The observed correlation for IPF cases will be compared to experimental results for outwardly propagating flames (OPFs). Effects of Flame Stretch The relationship between the the laminar burning velocity and the flame stretch was proposed by Markstein [12] to be Su ⳱ Suo ⳮ LK
(1)
Su is the local burning velocity, defined as the propagation velocity of the stretched flame with respect to the unburned reactants. Sou is the unstretched laminar burning velocity, and L is the Markstein length. The stretch rate K is defined as 1/A(dA/dt) [13] and describes the growth of an infinitesimal flame surface. The Markstein length is proportional to a characteristic flame thickness based on the mass diffusivity of the deficient reactant, dD, so that a dimensionless Markstein number can be defined as Ma ⳱ L/dD [14]. The stretch rate K can be non-dimensionalized as well by the characteristic flow time within the flame (residence time) to obtain the Karlovitz number Ka. This characteristic time has been defined by either using the unstretched [13] or stretched [14,15] values of the burning velocity and flame thickness. The
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Equation 4 has been shown [13] to be valid at low stretch rates and assumes a simple linear relation between the stretch and the burning velocity. Equation 5 is a nonlinear relationship between laminar burning velocity and the Karlovitz number introduced by Kwon et al. [15] to extend the applicability of equation 1 to finite levels of stretch [16]. Equation 5 has been shown to agree with experimental data for outwardly propagating spherical flames at low to intermediate stretch rates, for Ka* up to 0.7 [14,17]. One goal of this study is to determine if relations 4 or 5 agree with experimental data for IPFs. It is noted that both equations 4 and 5 give the same Markstein number at the limit of small stretch (K 0), when the stretched burning velocity approaches its unstretched value.
Stretch Rate of IPFs Fig. 1. Two theoretical predictions of the effects of flame curvature on the burning velocity (Su). Plot of equation 7 (—), which correlates the data of Kwon et al. [15] and equation 8 (– –), which is derived from asymptotic analysis of Law [13]. Both are calculated for a Markstein number of 7.0 which is the observed Markstein number for OPF with U ⳱ 0.65. Several contours of Ka* are included to give an idea to the amount of stretch imposed on the flame.
flame thickness (dD) has been defined as the ratio of mass diffusivity (Du) to the stretched burning velocity (Du/Su) [15] or as dD, which is the ratio Du/Sou [13]. In the case of combustion in air, the mixtures are highly diluted, so that Du is approximated as the binary diffusivity of the fuel with respect to the diluent (nitrogen), which equals 0.113 cm2/s for the mixture used [14]. Law [13] defines the Karlovitz number (Ka*) using the unstretched laminar burning velocity, as follows: Ka* ⳱
冤(SD ) 冥 K u o 2 u
(2)
Kwon et al. [15] used the stretched value of the burning velocity to define the following Karlovitz number:
冤 冥
Du Ka ⳱ K (Su)2
(3)
Two different relationships for laminar burning velocity that have been widely used to obtain Markstein numbers from experimental studies are: Su ⳱ 1 ⳮ MaKa* Sou
(4)
Su 1 ⳱ Sou 1 Ⳮ Ma Ka
(5)
For an ideal inwardly propagating spherical flame, the velocity of reactants inside the pocket is negligibly small because of symmetry requirements. The hydrodynamic strain rate is • u¯ ⳮ n¯ • (n¯ • )u¯ in the reactants just upstream of the flame; therefore, a spherical IPF is not subjected to hydrodynamic strain. Based on relations reported by Law [13], the stretch rate for such an IPF is K⳱
Su Rc
(6)
where Su is the local burning velocity and Rc is the radius of curvature of the flame surface. The radius of curvature is defined to be negative if the surface is concave toward the reactants, which occurs for IPF cases. Equation 6 can be combined with equations 2 through 5 to obtain two possible relations between the stretched burning velocity and the local flame curvature [13,15] Su 1 ⳱ Sou 1 Ⳮ Ma d oD/Rc
(7)
Su d oD o ⳱ 1 ⳮ Ma Su Rc
(8)
where d oD is the unstretched laminar flame thickness. Fig. 1 shows the differences between equations 7 and 8 for an assumed Markstein number of 7.0, which is the Markstein number extrapolated from OPF measurements for the current propane/air mixture (U ⳱ 0.65). The figure shows that equation 7 predicts a nonlinear increase of the local burning velocity as the pocket burns inward (for positive Ma numbers). However, equation 8 predicts that the velocity will increase linearly until the pocket is consumed. The present data are used to determine which of the two curves in Fig. 1 best represents the experimental observations.
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Fig. 2. Schematic of the combustion chamber showing pocket formation (left diagram) and the positions of the vortex and flame at an earlier time prior to the interaction (right diagram).
Experimental Setup The experiment that was used is shown in Fig. 2; a vortex-flame interaction creates a pocket of reactants in a premixed propane/air mixture at equivalence ratios (U) ranging from 0.58 to 0.65. Roberts and Driscoll [16] and Roberts et al. [17] report numerous images of pockets that were formed with this device. An upward moving laminar flame interacts with a downward moving toroidal vortex, causing the flame to wrap around the vortex. The distorted flame surface will eventually pinch off, so that a small bubble of unburned reactants will be completely surrounded by the flame surface. This pocket is studied long after the vortex has dissipated. The current setup consists of two rectangular chambers, which are joined together but initially separated by a removable splitter plate (Fig. 2). The lower chamber is where the interaction takes place. It contains an ignition source, exhaust ports, and quartz windows for optical access. The upper chamber is joined to a loudspeaker and contains a plate with a 4 cm diameter orifice (where the toroidal vortex is formed) and a pair of solenoid-operated pressure relief ports. By using two chambers, it is possible to separately control the equivalence ratio inside the vortex and in a region far outside the vortex. The mixture inside the vortex is kept very lean, while the mixture far from the vortex is a slightly richer propane/air mixture that will assist the flame to rapidly envelop the pocket formed by the vortex.
In this fashion, larger, slower burning pockets can be achieved without the use of strong vortices, which might not dissipate completely, which is a requirement of this experiment. The chambers are originally separated by a splitter plate, which is removed before the experiment is run. First, the exhaust port is opened, to prevent pressure buildup inside the chamber. A toroidal vortex is produced by driving the speaker with a rampvoltage generator. The vortex is formed in the upper chamber and will thus entrain only fluid contained in this section. The experiment is carefully timed so that the initial vortex/flame interaction occurs just after the vortex crosses the dividing line between both chambers. This measure reduces the amount of outside fluid entrained into the vortex. The difference in U between both chambers is also kept within 15% so that any small amount of mixing will have little effect on the equivalence ratio of the pocket. Using the vortex theory of Maxworthy [18], it was estimated that less than 20% of the total pocket volume is entrained from the outside gas mixture; therefore, the uncertainty in U is 3%. A spark creates a flame which travels upward through the bottom chamber. The pocket is initially tear shaped and has an initial diameter of 1 cm, and it becomes nearly spherical as its diameter decreases. Data Reduction Flame speeds were measured by recording shadowgraph images of the pocket burnup at 1500
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frames per second using a Fastex 16 mm high-speed movie camera. Images are digitized using a Kodak CCD camera with 2000 ⳯ 3000 pixel array. The digitized flame surface is fitted with a high-order polynomial. Fig. 3 shows several snapshots of the flame surface at different times. The local burning velocity (Su) was determined from Su ⳱ Dxn/Dt
Fig. 3. Flame surface at three times, showing the normal to the surface at two locations. U ⳱ 0.58.
Fig. 4. Measured burning velocities of IPFs for U ⳱ 0.62, Sou ⳱ 16.0 cm/s, and doD ⳱ 0.071 mm. Solid symbols represent results obtained using equation 9; open symbols were obtained using equation 11. The plot includes the predictions of equation 8 (– –) and equation 7 (– • –) utilizing a Markstein number of 7.3 that is interpolated from the data of Tseng et al. [14] for the present value of U. Several contours of Ka* were added to indicate the amount of stretch imposed on the flame.
(9)
where Dxn is the distance from one flame front to the next (Fig. 3) along the line normal to the initial surface, and Dt is the time between images [19]. The gas velocity inside the pocket is negligibly small, as discussed below. The local flame curvature and burning velocity are calculated at 400 points along the surface for each image. The curvature is defined as the average of the curvatures at the initial and final positions during flame travel. The three-dimensional radius of curvature Rc, is defined by [20] 1 1 1 ⳱ Ⳮ Rc R1 R2
(10)
where R1 is the component of curvature in a plane cutting through the center of the pocket (defined to be the inverse of the curvature of the two-dimensional flame surface). R2 is the out-of-plane component, which is the distance of the flame surface element to the axis of symmetry, Rsymm divided by cos h, where h is the angle between the normal vector and the axis of symmetry. Note that for a spherical flame both R1 and R2 are equal to the measured radius of the sphere (Rf), so Rc ⳱ Rf/2. It is important to ensure that at the time when measurements of the burning velocity are made, the vortex has attenuated such that the gas velocity inside the pocket is negligible with respect to the flame speed. The initial vortex will rapidly vanish due to the high viscosity caused by the flame and the dilation of the gas [3,21,22]. The velocity inside a typical pocket was measured using particle-imaging velocimetry (PIV) in a previous study [23]. A characteristic induced velocity Uchar was determined by dividing the total circulation inside the pocket (C) by 2pd, where d is the vortex diameter. From the PIV measurements, it was determined that at the time of pocket formation, only about 10% of the induced velocity remains and quickly dissipates before pocket burnout. The vortices used in the present work were very weak and had a characteristic initial vortex velocity of Uh 13 cm/s, which will decay to less than 1.3 cm/s at the moment of pocket formation. Since the burning velocities measured were 6.5 cm/s to 80 cm/s, the uncertainty in flame speed measurement that arises from a non-zero reactant velocity varies from 1.6% to 20%.
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Results
Fig. 5. Measured burning velocities of IPFs for U ⳱ 0.65, Sou ⳱ 18.6 cm/s and doD ⳱ 0.061 mm. Solid symbols represent results obtained using equation 9; open symbols were obtained using equation 11. The plot includes the predictions of equation 8 (– –) and equation 7 (– • –) utilizing a Markstein number of 7.0 that is interpolated from the data of Tseng et al. [14] for the present value of U. Several contours of Ka* were added to indicate the amount of stretch imposed on the flame.
Figure 4 shows results for the U ⳱ 0.62 flame; the non-dimensional burning velocity is plotted against the non-dimensional flame curvature for the last eight frames of the interaction. If the relation used by Kwon et al. (equation 8) to correlate their OPF data applies to the IPF, the data points in Fig. 4 would lie on a line that passes through Su/Sou ⳱ 1 and would have a slope equal to Ma. In Fig. 4, about 2000 points are plotted using a moving-average scatter plot. The scatter, shown by the standard deviation brackets, is significant due to measurement uncertainties discussed later, but basic trends can be deduced. The lower dotted line in Fig. 4 represents equation 8 for a Markstein number of 7.3, which is the value extrapolated from data obtained Tseng et al. [14] for an OPF. The upper dotted curve in Fig. 4 is equation 7 for the same Markstein number. The IPF data points do not match either of these relations but instead follow a straight line (in agreement with equation 8) with a slope of 22.8, which is the Markstein number for the IPF at U ⳱ 0.62. It is important to note that Fig. 4 shows that there is a large increase of the stretched burning velocity in the present experiment, which is as large as six times the unstretched value. A second method to determine the burning velocities was used, which involved the volume, V, and surface area, A, of the flame during the final phase of pocket burnout. During the last few frames, the flame pocket is nearly spherical, so that a global value of Su and Rc can be obtained using the following equations (valid only for a sphere):
冢
Fig. 6. Unstretched laminar burning velocity of propane/air flames obtained by Law et al. [24] and Vagelopolous et al. [25]. The observed extinction conditions in the present experiment (䉴) are included as reference.
1/3
冣
dV 1 1 3 Su ⳱ ⳮ and Rc ⳱ ⳮ V dt A 2 4p
(11)
The volume, V, was determined from a numerical integration using the surface boundary (Fig. 3) and assuming axisymmetry. Results for this second method are shown as triangles in Fig. 4, and produce final Markstein numbers that agree with values obtained using equation 9 to within 10%. Fig. 5 is an identical plot showing the results for U ⳱ 0.65; the experimental Ma of 19.3 is considerably larger than the value of 7.0 determined from OPF experiments. The data in Figs. 4 and 5 show that the present results are in good agreement with the linear relation given by equation 8; however, the Markstein number for an IPF is measured to be two to three times larger than that of a corresponding OPF. Fig. 6 is a plot of the unstretched laminar burning velocity, Sou versus equivalence ratio that was obtained from data by Law [24] and Vagelopolous et al. [25]. The values of Sou needed to normalize the measurements for equivalence ratio of U ⳱ 0.58 were obtained by extrapolating the data from Fig. 6.
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Fig. 7. Comparison of experimentally obtained Markstein numbers for present IPFs with Markstein numbers obtained from several OPF experiments and numerical simulations of Tseng et al. [14], Taylor et al. [26], and Sun et al. [4]. Each open circle represents the average of 2000 flame segments obtained from one run (such as in Fig. 5); solid circles are the average of multiple runs.
Comparisons to Other Negatively Stretched Flames Figure 7 shows how the values of Markstein number deduced for the IPFs (using equation 8) compare with values reported for the OPFs studied by Tseng et al. [14], Taylor et al. [26], and Sun et al. [4]. It is seen that the present values of Ma for IPF cases are two to three times larger than those reported for the OPFs. While the disagreement between the IPF and OPF cases in Fig. 7 cannot yet be adequately explained, there have been several other studies which have reported larger values of Ma for an negatively stretched flame than the reported values for a corresponding OPF. Echekki and Mungal [9] measured velocities near the tip of a Bunsen flame and reported that the slope of their measured curve of Su versus flame curvature is 6.46 for a stoichiometric methane/air mixture, which yields a value of Ma of 6.46, based on equation 7. Tseng et al. [27] report that the Markstein number for their OPF in this same mixture is 1.3. Thus, the Markstein number of Echekki and Mungal’s IPF is five times the OPF value reported by Tseng et al. for the same fuel/air mixture. In the numerical study of Echekki and Chen [3], the propagation speeds of curved flames were computed using a reduced four-step mechanism. They reported a curve similar to Fig. 4 in which the propagation speed was plotted against the flame curvature; they found that the slope of this
curve was more than three times larger in the negative curvature region than in the positive curvature region. In a related computational study of curved flames, Chen and Echekki [28] showed that near a cusp (where the curvature is negative), their computed local propagation speeds were more than 50 times the unstretched value (which will tend to result in large Markstein numbers). Negative curvature was shown by Chen and Echekki to focus certain light radicals, such as H, O, and OH, that are important in the chain-carrying reactions, which will result in increased local heat release and larger burning velocities. Numerical studies by Sun et al. [4] showed good agreement between OPFs and IPFs at very low stretch rates. It is not yet clear whether the differences between their Markstein numbers and the present values are due to finite stretch rates, preheating of gas inside the pocket, or complex chemistry.
Experimental Uncertainties and Effects of Flame Thickness A large amount of experimental uncertainty is observed in Figs. 4 and 5, as noted by the large error bars, which denote the root-mean-squared deviations of the data when 2000 flame segments are considered. Most of this uncertainty is due to properly aligning the individual images to obtain the overlaid image such as the one shown in Fig. 3. The pocket of reactants moves downward, carried by the bulk of the exhaust, as the flame burns inwardly. To remove the effects of this convection, the images had to be moved along the axis of symmetry to obtain Fig. 3. It is noted that there is no alignment problem when equation 11 was used to determine the triangle symbols in Figs. 4 and 5, since equation 11 only requires information about the volume and surface area of the pocket and not its location relative to other images. However, equation 11 could only be used in the late stages of pocket burnup when the pocket is nearly spherical. Additional uncertainties are introduced because the flame surface is fitted with a polynomial and because the flame boundary has a finite thickness on the digitized shadowgraph images. The flame boundary is defined as the line between the white line and the black line that appears on the shadowgraph; this line is the location of the maximum value of the first derivative of the gas density. The initial image has a viewing area of 3.6 ⳯ 2.4 cm and is digitized by a 3000 ⳯ 2000 pixel array, which provided a spatial digitization uncertainty of 3.6 cm/ 3000, which is 12 lm and is too small to be a source of error. The uncertainty associated with identifying the flame boundary, due to shadowgraph limitations
STRETCHED INWARDLY—PROPAGATING PREMIXED FLAMES
and curve fitting the surface, is approximately 4 pixels or about 48 lm. The typical displacement of the flame segment is 0.437 mm in Fig. 3. So the ratio of spatial uncertainty to displacement is about 10% and is not a significant contributor to the much larger uncertainties shown in Fig. 4. The ratio of the flame thickness to the pocket diameter (dD/df) for the current experiment ranges from 0.002 to 0.025. Thus, the flame thickness (Du/ Su) is always less than 3% of the pocket diameter. The flame thickness decreases by a factor of ten as the pocket becomes smaller for the present conditions, because the flame speed increases by a factor of ten, as was shown in Fig. 4. Not all flames behave this way; for fuel/air combinations that have a negative Markstein number, the flames would not become thinner as the pocket becomes smaller. If the ratio (dD/df) approaches unity (which it does not in the present experiment), then the curvature of the leading edge of the flame differs from that of the trailing edge, and the unsteady accumulation of mass inside the flame must be considered when writing the equation for the conservation of mass, as discussed by Sun et al. [4].
Conclusions 1. The present experiment is shown to provide a useful way to create a repeatable and controllable IPF which is tear shaped initially and becomes nearly spherical in the later stages. IPFs provide a way to measure the effects of negative flame curvature accurately and without the added complexity of hydrodynamic strain. 2. The burning velocity–curvature correlation follows a linear trend, in agreement with equation 8 adequately explains the results; however, the slope of the correlation curve yields a Markstein number that is two to three times larger than the Markstein number reported previously for OPFs in the same fuel/air mixture. 3. The reasons why the Markstein number is different for the IPFs and OPFs is not known, but the present findings are consistent with previous studies of Bunsen tips and with numerical simulations which have reported similar trends. Acknowledgments Support for this research was provided by National Science Foundation grants CTS 9123834 and CTS 9904198, administered by Dr. Farley Fisher, and NASA grant NCC3656.
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REFERENCES 1. Peters, N., Proc. Combust. Inst. 21:102:1231–1250 (1986). 2. Lee, T.-W., North, G. L., and Santavica, D. A., Combust. Flame 93:445–456 (1993). 3. Echekki, T., and Chen, J. H., Combust. Flame 106:184–202 (1996). 4. Sun, C. J., Sung, C. J., He, L., and Law, C. K., Combust. Flame 118:108–128 (1999). 5. Sun, C. J., and Law, C. K., Proc. Combust. Inst. 27:963–970 (1998). 6. Chen, J. H., and Im, H. G., Proc. Combust. Inst. 27:711–717 (1998). 7. Haworth, D., and Poinsot, T., J. Fluid Mech. 244:405– 436 (1992). 8. Bradley, D., Gaskell, P., and Gu, X., Combust. Flame 104:176–198 (1996). 9. Echekki, T., and Mungal, M. G., Proc. Combust. Inst. 23:455–461 (1990). 10. Poinsot, T., Echekki, T., and Mungal, M. G., Combust. Sci. Technol. 81:45–73 (1992). 11. Wagner, T. C., Combust. Flame 59:267–272 (1985). 12. Markstein, G. H., J. Aeronaut. Sci. 18:199–209 (1951). 13. Law, C. K., Proc. Combust. Inst. 22:1381–1402 (1988). 14. Tseng, L.-K., Ismail, M. A., and Faeth, G. M., Combust. Flame 95:410–426 (1993). 15. Kwon, S., Tseng, I.-K., and Faeth, G. M., Combust. Flame 90:230–246 (1992). 16. Roberts, W. L., and Driscoll, J. F., Combust. Flame 87:245–256 (1991). 17. Roberts, W. L., Driscoll, J. F., Drake, M. C., and Goss, L. P., Combust. Flame 94:58–69 (1993). 18. Maxworthy, T., J. Fluid Mech. 51:15–32 (1971). 19. Sinibaldi, J. O., Mueller, C. J., and Driscoll, J. F., Proc. Combust. Inst. 27:827–832 (1998). 20. Matalon, M., Combust. Sci. Technol. 31:169–181 (1991). 21. Poinsot, T., Veynante, D., and Candel, S., J. Fluid Mech. 228:561–606 (1991). 22. Mueller, C. J., “Measurements of a Flame-Vortex Interaction Dynamics and Chemistry,” Ph.D. thesis, University of Michigan (1996). 23. Mueller, C. J., and Driscoll, J. F., Proc. Combust. Inst. 26:347–355 (1996). 24. Law, C. K., Zhu, D. L., and Yu, G., Proc. Combust. Inst. 21:1419–1426 (1998). 25. Vagelopolous, C. M., Egolfopoulus, F. N., and Law, C. K., Proc. Combust. Inst. 25:1341–1345 (1994). 26. Taylor, S. C., Ph.D. thesis, University of Leeds, U.K., 1991. 27. Aung, K., Tseng, L.-K., Ismail, M., and Faeth, G. M., Combust. Flame 102:526–530 (1995). 28. Chen, J. H., and Echekki, T., Combust. Flame 116:15– 48 (1999).
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COMMENTS Stephen Klotz, Princeton University, USA. You described how you account for internal circulation (caused by the vortex generation process). Have you accounted for the relative motion of your flame product and the surrounding gas in your chamber? If so, how does this flow affect your flame speed measurements? Have you considered using PIV (or a similar technique) to determine the important fluid motions in your flame generation device?
decreased to 70 mm. So the range of the present measurements, the ratio of the thermal thickness to the flame diameter varies from 0.09 to 0.21. These numbers indicate that little, if any, overlap of preheat zones occur when the measurements were made.
Author’s Reply. The flame propagation speed is defined as the speed of the combustion wave with respect to the reactants. Therefore, the relative motion of the flame pocket and the surrounding gas will not have any affect on the propagation speed, unless the surrounding gas somehow causes a relative motion between the reactants inside the pocket and the thin flame zone. As the pocket moves out of the chamber, both the pocket of reactants and the surrounding flame are convected at the mean flow velocity. Any velocity field that is uniformly superimposed on both the thin flame zone and the reactants will, by definition, have no effect on the propagation speed. There is no physical reason (and we have no evidence) to suggest that the vertical convection speed of the flame should differ from that of the pocket of reactants. The fact that our measured propagating speeds approach the known value of the unstretched propagation speed for small values of stretch indicates that the measurement procedure is sound.
1. Bechtel, J. H., Blint, R. J., Dasch, C. J., and Weinberger, D. A., Combust. Flame 42:197–213 (1981).
● Jacqueline H. Chen, Sandia National Laboratories, USA. As the flame burns inward, at some point the enhancement of the flame propagation is influenced by mutual annihilation. Can you quantify the importance of this effect and at what radius of curvature it becomes significant? Author’s Reply. Yes, as two segments on opposite sides of the spherical flame approach each other, their preheat zones eventually will overlap, which can greatly increase the propagation speeds of each segment. However, the present measurements were made before this overlap begins. We note that this overlap is fundamental property that possibly can occur in all flames with negative curvature, including inwardly propagating flames, bunsen-tip flames, and certain segments of turbulent flames. It may explain why the present results differ from those reported for outwardly propagating flames. The thermal thickness of the preheat zone is Tmax ⳮ Tmin)/(T/x)max which is measured to be 7.4 ␣/SL in laminar flames [1]. Based on this estimate, the thermal thickness of the flame described in Fig. 5 is 0.30 mm when the flame diameter is 3.40 mm (when the earliest measurements are made). At a later time when the final measurements are made, the estimated thermal thickness has decreased to 0.15 mm and the flame diameter has
REFERENCE
● Yiguang Ju, Princeton University, USA. Since laminar flame speed depends on the local velocity normal to the flame front on the unburned side and on the local flame structure, how do you make sure that the flow in the unburned pocket is zero and the structure of the fine evoluting local flame structure is the same as the ideal inward propagating flame? Author’s Reply. Steps were taken to ensure that the vortex inside the pocket has decayed sufficiently such that any gas velocity in the reactants is less than 10% of the flame propagation speed and therefore can be neglected in determining the propagation speed. First, a sufficiently weak vortex was chosen such that it will decay to negligible strength before the pocket is formed. Secondly, the time for the vortex to decay completely. Thirdly, PIV measurements from a previous study [1] were used to quantify the velocity of reactants inside the pocket. Based on the previous PIV measurements of [1], the maximum possible error introduced by reactant motion is 1.4 cm/s. The measured propagation speeds in Fig. 5 vary from 32 cm/s to 64 cm/s, so this error introduced by reactant motion varies from 2% to 4%.
REFERENCE 1. Mueller, C. J., Driscoll, J. F., Reuss, D. L., Drake, M. C., and Rosalik, M. E., Combust. Flame 112:342–358 (1998). ● William Roberts, North Carolina State University, USA. How do you ensure the apparatus is isobaric? If it is not, then you would have compression of the flame pocket, leading to a higher apparent SL. Could this explain your higher Ma number? One of your possible explanations for higher Ma number is preheating of the pockets. If this were true, the pocket would expand, leading to a lower apparent
STRETCHED INWARDLY—PROPAGATING PREMIXED FLAMES SL. Can you give a weighting of SL increase due to increasing To and SL decrease due to expansion? Author’s Reply. 1. The pressure inside the chamber does not increase by more than approximately 10% of its initial value of 1 atm. There are large exhaust ports which remain open at one end of the chamber, and there are pressure relief valves which are open at the other end of the chamber. The very lean equivalence ratios used (0.55 to 0.6) and the slow movement of the flame (10 cm/s) also prevent significant pressure rise. 2. The present flames become thinner as they burn inward, which would tend to prevent merging of the preheat zones and reduce any preheating of the reactants. However, it is not known if preheating of the reactants does occur. If the temperature of the reactants is increased by 60 C, for example, this would approximately double the unstretched burning velocity but would only increase the radius of the pocket by 6%. Therefore, the effects of gas density changes are negligible compared to the sensitivity of the flame speed to the temperature of the reactants. We note that if the reactants are being
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heated, this would be a fundamental characteristic of inwardly propagating flames, and may explain why such flames may have different properties than outwardlypropagating flames. Such heating, if it occurs, would not affect the accuracy of the present measurements or the generality of the results for IPFs. ● Arvind Atreya, University of Michigan, USA. This is excellent experimental work. I wonder if you have studied fuels other than propane to see the Lewis number effect? Author’s Reply. No, we have not. It is difficult to achieve a nearly spherical flame speed as the pocket of fuel burns out. Lean propane-air flames become spherical in this experiment because they are diffusionally stable and do not have small scale wrinkles. Increasing the equivalence ratio creates less stable flames which have small wrinkles, which cause a non-spherical pocket. Future work is planned to study lean methane-air flames, but since they are diffusionally unstable and wrinkle easily, it may be difficult to achieve a final flame shape that is spherical.