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Measurements of flamelet orientations in premixed flames with positive and negative Markstein numbers
Proceedings of the Combustion Institute, Volume 28, 2000/pp. 367–373
MEASUREMENTS OF FLAMELET ORIENTATIONS IN PREMIXED FLAMES WITH POSITIVE AND NEGATIVE MARKSTEIN NUMBERS D. A. KNAUS and F. C. GOULDIN Sibley School of Mechanical and Aerospace Engineering Cornell University Ithaca, NY 14853, USA
Measurements of instantaneous flamelet surface orientation of preferential diffusion stable, ethylene/ air and unstable, methane/air flames are presented for three different values of normalized turbulence intensity, u/SoL , where SoL is the unstretched laminar flame speed. Instantaneous flamelet orientations were measured in three dimensions via crossed-plane imaging. Repeated measurements were used to estimate probability density functions (PDFs) of flamelet orientation. A one parameter fit of the PDF of flamelet orientations was applied that has been applied previously to data from V-flames and from spark ignition engine flames. The fits of the PDF were in good agreement with measurements. The fit parameter, f, which describes the breadth of the distribution of flamelet orientations, was found to be systematically larger for stable flames than for unstable flames at similar values of u/SoL . The degree of wrinkling of the ¯ averaged across the flame brush, and flamelet isdescribed quantitatively by the flamelet surface density, R, did not vary for stable and unstable flames, contrary to expectation. f data from the current work are in good agreement with data from previous experiments. A 具c典 constant contour image is presented to give an indication of the shape and width of the turbulent flame brush. Normals to 具c典 constant surfaces, N具c典, were measured for the first time in V-flames and were found to be colinear with the mean of the flamelet normals, 具N典.
For steady, laminar flames with small Karlovitz number, Ka, the effect of stretch on the flame speed can be described by the equation [2]
Introduction Premixed turbulent flames in the flamelet regime are characterized by regions of reactants and regions of products separated by a thin flame sheet, or flamelet. The primary assumption of the flamelet description is that a surface element on the flamelet has the structure of a perturbed laminar flame and is propagating locally with respect to the reactants at the corresponding laminar burning velocity modified to account for the local perturbation of the flamelet element. The burning intensity is related to the degree of wrinkling of the flamelet, which increases the flamelet surface area per unit volume and therefore increases the rate of reactant consumption per unit volume. For the flamelet regime, the mean rate of creation of products, 具w典, can be written 具w典 qrSLo IoR reactant density, SoL
(1)
where qr is the is the unstretched laminar flame speed, Io is a factor that accounts in the mean for perturbation effects on SoL , and R is the mean flamelet surface area to volume ratio, or the flamelet surface density [1]. When written this way, information about the degree of wrinkling of the flamelet is contained in R, and information about the effect of the wrinkling on the local flame speed is contained in Io.
SL [1 MaKa]1 SoL
(2)
In equation 2, SL is the flame speed and Ma is the Markstein number. Ka is a dimensionless number quantifying the stretch magnitude, and Ma is a dimensionless number containing the Markstein Length, L. The value of L and in turn Ma determines the effect on the flame speed of the flamelet stretch. For positive Ma flames, the flame speed decreases with positive stretch, and for negative Ma flames, the flame speed increases with positive stretch. For a positive Ma flame, a surface element with positive curvature, such as that at the tip of a finger of products protruding into a reactant region, is slowed down, and a surface element with negative curvature, such as that at the tip of a finger of reactants protruding into a product region, speeds up. This is referred to as a stable preferential diffusion condition, because a perturbation of the flamelet surface (i.e., a wrinkle) is damped [3]. An unstable preferential diffusion condition exists for negative Ma flames. A surface element with positive curvature, such as that located at the tip of a finger of products protruding into a reactant region, speeds
367
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up, and a surface element with negative curvature, such as that located at the tip of a finger of reactants protruding into a product region, slows down. The result is an amplification of the wrinkling of the flamelet. Positive Ma flames are preferential diffusion stable, and negative Ma flames are preferential diffusion unstable [3]. This effect is referred to as preferential diffusion because the value of Ma for a flame is dependent on, among other things, the relative mass diffusivities of the fuel and oxidizer. For flames where the fuel and oxidizer have significantly different diffusivities, fuel and oxidizer approach the flame zone at different rates due to stretch, which affects the flame speed [3]. Ma describes a flame’s response to stretch and is therefore dependent on the relative diffusivities of fuel and oxidizer. Numerous workers have provided evidence of the importance of Ma and Le, Lewis number, effects on the surface topography, and burning rate of premixed flames. Wu et al. [4] presented experimental results indicating variations in flamelet wrinkling for different Ma turbulent flames. Bradley [5] presented data that indicate the need to correlate experimental values of turbulent burning velocity with KaLe. He also stated that it is ultimately more logical to correlate with KaMa, pending the availability of Ma data, which are becoming more and more available. Haworth and Poinsot [6] demonstrated the importance of Le effects in premixed turbulent flames using numerical calculations, as did Echekki and Chen [7]. The primary purpose of the current work was to observe and quantify the degree of wrinkling of positive and negative Ma flames. Two flames were studied, one with a positive Ma, and one with a negative Ma, at various values of u/SoL, where u is the turbulence intensity in the reactants approaching the flame. Flamelet orientation measurements were made via crossed-plane imaging [8–11]. Crossedplane imaging allowed the instantaneous flamelet surface normal vector, N, to be measured in three dimensions via two simultaneous laser tomography measurements made in orthogonal planes. The two tomography laser sheets intersected along a line referred to as the measurement line. At points where the measurement line intersected the flamelet the orientation of the flamelet surface within the plane of each laser sheet was measured. Because the orientations of the laser sheets in space were known, N could be measured in three dimensions. N was defined as a unit vector normal to the flamelet surface oriented in the direction of flamelet propagation, that is, toward reactants. The probability density function (PDF) of N was estimated from repeated measurements and used to help quantitatively describe the degree of wrinkling of the flamelet. It was expected that a positive Ma number (stable) flame would exhibit a lower degree of wrinkling than a
negative Ma (unstable) flame for the same value of u/SoL. Specifically, N data were obtained for three stable ethylene/air flames and three unstable methane/air flames. These data were used to test a single parameter PDF fit that was previously applied to V-flame [8] and spark ignition engine (SI) flame data [11] and is of the form P(,h)sin ddh A|N • ny|exp (2/f2) sin ddh
(3)
Here N is expressed in terms of spherical coordinates and h, and A is a normalization constant. The vector dot product term accounts for crossingweighting in the measured PDFs. Crossed-plane imaging measurements are inherently crossingweighted: the probability associated with a flamelet of a certain orientation being measured is weighted by the probability that a flamelet having that orientation crosses the measurement line [8]. ny is a unit vector aligned with the measurement line. The following expression relates N data to R[12]: R
冬|r |冭 1
g
nc
(4)
c
The bracket denotes an ensemble averaged quantity. rg is the direction cosine of N and a line g, which is defined as spanning the turbulent flame brush normal to mean progress variable, 具c典, constant surfaces. nc is the flamelet crossing density, the average number of flamelet crossings of g per unit length. The c subscript indicates that the average is crossingweighted. The apparent singularity in equation 4 for rg 0 is not realized due to the crossing-weighting of the average, since a surface parallel to a line does not cross the line [8,9]. R describes the average degree of wrinkling of the flamelet at a point in space. The average degree of wrinkling across g is quantified by the expression ¢ 1 R lb
⬁
冮
⬁
Rdg
(5)
where lb is the flame brush thickness. The instantaneous progress variable, c, describes the progress of reaction of premixed flames. It is defined to be 0 in reactants and 1 in products. For flames in the flamelet regime, the reaction zone can be approximated as infinitesimally thin, and the instantaneous c field is bivalued, 0 in reactants and 1 in products. The mean progress variable, 具c典, field describes the shape and width of the turbulent flame brush. The 具c典 field can be measured within the planes of the two laser sheets used for crossed-plane imaging. Images are made binary to obtain instantaneous c fields and then averaged. The flame brush thickness, lb, is defined as
FLAMELET ORIENTATIONS IN DIFFERENT MARKSTEIN-NUMBER FLAMES
369
TABLE 1 Flame conditions and summary of results Flame Methane 1 Methane 2 Methane 3 Ethylene 1 Ethylene 2 Ethylene 3
U
SoL (cm/s)
u/SoL
Ma
f (⬚)
具1/|rg|典c
lb (mm)
¯ (mm1) R
0.65 0.65 0.65 0.61 0.61 0.61
15 15 15 26 26 26
0.60 0.90 1.16 0.51 0.77 1.00
0.14 0.14 0.14 2.77 2.77 2.77
28 30 30 30 35 36
1.12 1.14 1.14 1.14 1.20 1.21
0.93 0.81 0.78 0.94 0.91 1.11
1.20 1.41 1.46 1.22 1.32 1.09
Fig. 1. SoL and Ma as a function of equivalence ratio for methane-air (closed symbols) and ethylene-air (open) flames. The SoL curves are fits of the data found in Law [14]. The Ma curves are based on correlations found in Aung et al. [15].
lb
⬁
冮
⬁
具c典 (1具c典) dg
(6)
and can be evaluated from 具c典 field image data by integration. From crossed-plane 具c典 field images, normals to 具c典 constant surfaces, N具c典, can be measured along the measurement line in a way analogous to that for finding N. In previous crossed-plane imaging measurements of turbulent methane/air flames, it was found that the surface-weighted PDF of N is uniform in the azimuthal angle when written in spherical coordinates with the polar axis aligned with 具N典 [8], while N具c典 and 具N典 were found to be colinear for the SI engine flames studied in Ref. [11]. A secondary purpose of the current work was to determine whether 具N典 and N具c典 are colinear for V-flames. Experiment Experiments were performed on the V-flame burner used in several previous studies in this laboratory [8–10,13,14]. Fuel and air flow rates were measured with mass flow meters. A portion of the air flow was diverted through a blast-atomizer-type
seed particle generator, within which it was seeded with silicone oil droplets, and then passed through a cyclone separator. Droplet sizes were estimated to be no more than a few micrometers [13]. The reactants were mixed in a plenum at the base of the burner. A square mesh wire grid with approximately 4.5 squares/cm2 and a 1 mm wire diameter were positioned 50 mm upstream of the burner exit. Measurements were made at an axial distance of 40 mm downstream of the stabilizer rod, which was located at the burner exit. The flames studied are described in Table 1; they were chosen so that measurements could be made on both positive and negative Ma flames over similar ranges of u/SoL. Lean flames are preferred to avoid soot formation, which can interfere with laser tomography measurements. Values of SoL are taken from Ref. [14]. Linear correlations relating the Ma to the equivalence ratio, U, for methane/air and ethylene/air flames found in Aung et al. [15] were used to estimate Ma. For sufficiently lean mixtures, ethylene/air flames have positive Ma values and methane/air flames have negative Ma values, as can be seen in the plots of Ma and SoL in Fig. 1. For each fuel studied, a fixed value of U was chosen so that for each fuel Ma is fixed. u was then varied, by varying the flow rate, so that flames with similar values of u/SoL were measured for each fuel. u was determined from a linear fit of u to the area averaged mean jet velocity based on previous laser doppler velocimetry measurements performed on the burner [8]. Values of Ka for the flames studied were estimated to be on the order of 106 and were small enough that equation 2 was expected to apply to these flames [2]. The frequency-doubled 532 nm beam from a Nd:YAG pulsed laser (7 ns pulse duration) was used for tomographic visualization. The 6.4 mm diameter beam was focused and split into two mutually orthogonal light sheets approximately 70 mm wide, with waist thicknesses of less than 0.5 mm. The polarization of each beam was rotated via half-wave plates so that it laid in the plane of its light sheet. The two sheets were positioned so that each crossed over the burner at a 45⬚ angle to the vertical, with
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Fig. 2. Schematic representation of crossed-plane imaging measurements on a turbulent V-flame. The light from orthogonal laser sheets is scattered by oil droplets seeded in the reactants.
Fig. 3. 具c典 regions for flame Methane 1. Also shown is the position of the measurement line. The 具c典 domain, from 0 to 1, is divided into tenths, each represented by a grayscale color. The boundary of color regions represent the location of 具c典 constant contours.
their line of intersection perpendicular to the stabilizing rod. Fig. 2 shows schematically the physical appearance of the laser sheets interacting with the V-flame and the laboratory coordinate system with the y axis aligned with the measurement line, the z axis vertical, and the x axis aligned with the stabilizer rod such that the coordinate system is right-handed. Tomographic images were recorded simultaneously with two digital cameras, each having 256 color grayscale, 652 494 pixel CCD arrays. The cameras were positioned perpendicular to the laser sheets; that is, one sheet was parallel to the camera image plane, and the second was edge-on for each camera. They were fitted with 50 mm focal length lenses and 10 mm extension tubes, which provided a field of view of approximately 18 mm 24 mm. The camera resolution was greater than 0.2 mm. Polarizing filters mounted on the end of the lenses
blocked scattered light from the edge-on sheet. Hence, each sheet was imaged individually, with negligible interference from the light scattered from the other [8]. Ten nanometer bandwidth, laser line filters were used to reduce the amount of flame radiation reaching the cameras. The external trigger of the laser was used to trigger the cameras, and the camera exposure times were set to the minimum value, 10 ms. Oil droplets scattered the laser light in the reactants but were vaporized by the flame. The images were therefore bright in the reactants and dark in the products; the boundary between the two regions represents the curve of the flamelet surface within the plane of the laser sheet, the flamelet boundary. Images were recorded and saved for postprocessing. Image processing consisted of a thresholding step, which made the images binary, followed by the binary image operations of opening and closing [16] that filtered island features of diameters less than the opening/closing diameter, which was set at 5 pixels or ⬃0.2 mm. Flamelet crossings were identified in each image as points where the flamelet boundary intersected the measurement line. The flamelet boundary was found by an edge-finding algorithm and then fitted over a fit-width of 100 pixels at each crossing location. From a fit of the flamelet, its slope was determined at flamelet crossing points and used to define tangent vectors to the flamelet boundary. The tangent vectors in orthogonal image pairs for a given flamelet crossing were orthogonal to each other and tangent to the flamelet surface. The normalized cross-product of these tangent vectors was N. Further details on image processing can be found in Refs. [8] and [10]. For each flame studied, 1000 crossed-plane image pairs were recorded. The 具c典 fields in the planes of the two laser sheets were measured from the recorded crossed-plane images by averaging the images after they were made binary. Fig. 3 shows different 具c典 regions for flame Methane 1. The 具c典 values are grouped in tenths, each represented by a grayscale color. The boundaries between grayscale regions in the image represent 具c典 constant contours, the intersections of 具c典 surfaces with the plane of the laser sheet. The normals to 具c典 constant surfaces can be measured the same way N is measured: tangent vectors to a 具c典 constant contour are calculated from a fit of the 具c典 constant contour for both 具c典 field images, and the normalized cross-product of the tangent vectors is N具c典.
Results and Discussion For turbulent V-flames, it is expected that 具c典 surfaces are planes, and in the y-z (vertical) plane, it is expected that 具c典 surfaces appear as straight rays
FLAMELET ORIENTATIONS IN DIFFERENT MARKSTEIN-NUMBER FLAMES
371
Fig. 4. Projections onto the x-z plane of the tips of N具c典, 具c典0.5, and of 具N典 vectors for the flames studied.
originating from the stabilizer rod. This expected behavior can be observed qualitatively in Fig. 3, although the 具c典 constant contours are not perfectly straight. Note that the image plane shown in Fig. 3 is not coincident with the y-z plane but is 45⬚ from the vertical. Fig. 4 shows the projections of the end points of N具c典, 具c典 0.5, onto the x-z plane for all the flames studied. It is expected that N具c典 will have a small negative z component (down in the laboratory coordinate system) due to the shape of a V-flame and no x component. In Fig. 4, the data for N具c典 all have a small positive x component. Values of the angle that the projection of N具c典 onto the x-z plane make with the z axis are in the range of 6–16⬚. N具c典 vectors not being in the vertical plane is attributable to two possible causes. First, there is uncertainty in the orientation of the stabilizing rod with respect to the measurement line, which is estimated to be 5⬚. Second, there may be some non-uniformity in the mean axial velocity profiles exiting the burner. Although N具c典 are deviating slightly from the expected orientation, the important conclusion is that N具c典 are aligned with 具N典 to within a small difference, also shown in Fig. 4. To present the N data, we defined a spherical coordinate system with a polar angle, ; an azimuthal angle, h; and its polar axis aligned with 具N典. Because 具N典 is aligned with N具c典, either vector could be used to define the pole of the spherical coordinate system. Instantaneous flamelet normals were measured over a range of 具c典 along the measurement line. Due to the limited number of flamelet normals measured, it was not possible to obtain statistically valid estimates of N data PDFs for narrow 具c典 ranges. Instead, all measured normals for a flame condition were
Fig. 5. Measured marginal PDFs in (top) and h (bottom) for flame Ethylene 2. Also shown are fits based on a fit of the joint PDF of the form P(,h)A[N•ny]exp(2/ f2). For the flame shown, f35⬚.
grouped into one distribution. For SI engine flames, the PDFs of flamelet normals were found to vary only slightly over the range 0.1 ⬍ 具c典 ⬍ 0.75 [11]. Intuitively, one would expect that the distribution of flamelet orientations will broaden in the extreme 具c典 regions, 具c典 → 0 or 1. This expected broadening of N is expected to have a only a small effect on the crossing-weighted PDF of N and on R, since for 具c典 → 0 or 1, nc is small; that is, relatively few normals from the flame brush edges are measured and therefore contribute to the PDF estimate or to R calculations. Distributions of flamelet orientations can be presented in the form of marginal PDFs in and h, P() and P(h), which are shown in Fig. 5a and b for the ethylene 2 flame. P() is formed by generating a histogram of measured values, independent of h, and normalizing. P(h) is generated by forming a histogram of h values independent of and normalizing. Also shown in Fig. 5a and b are marginal PDFs based on a fit of the joint PDF, equation 3. The fit and data are in good agreement for ethylene 2 and for all of the flames studied. This demonstrates that the form of the PDF fit applies to ethylene flames as well as methane flames. Much of the scatter in the PDF data can be attributed to uncertainty associated with the number of samples in each histogram bin. An error bar is shown in Fig. 5a and b
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Fig. 6. PDF fit parameter, f, and average flamelet sur¯ plotted as a function face density across the flame brush, R, of normalized turbulence intensity. Also shown are previous f values for methane-air flames [8] measured at the same axial distance from the V-flame stabilizing rod as the current work. The lines represent linear fits of f values for the current ethylene flames and for both the current and previous methane flames combined.
for uncertainty due to bin sample size. Removing the crossing-weighting term, |N•ny|, from equation 3 gives the corresponding fit of the surface-weighted [8,9] joint PDF. When the spherical coordinate system is defined with the pole aligned with 具N典, or equivalently N具c典, the surface-weighted joint PDF is independent of h, and the distribution of N is described by the surface-weighted marginal PDF in alone, which is quantified in turn by the fit parameter, f. Flames with narrow distributions of N have narrow, peaked marginal PDFs in , and small values of f. Conversely, flames with broad distributions of N have broad marginal PDFs in , and large values of f. Values of f for the flames studied are given in Table 1 and plotted as a function of u/SoL in Fig. 6. Also presented in Fig. 6 are data for methane/air flames from previous measurements [8]. Inspection of the linear fits of the data show that f values for the ethylene flames are systematically higher than those for the methane flames, the opposite of the expected behavior. Comparison of the current data and the previous data give an indication of the repeatability of the experiment. The degree of wrinkling of the flames studied is ¯ and can be determined from equaquantified by R tion 5. A single value of f is assumed for all g, so that the mean inverse direction cosine term can be factored out of the integral. Convolution of the fit of the joint PDF with the inverse direction cosine term in equation 4 yields the mean inverse direction cosine term for a given f value (Table 1). Also shown in Table 1 are values of the flame brush thickness, lb. For the flames studied, the integral of nc across g, or the average number of flamelet crossings of g, ¯ 具1/|rg|典c/lb. R ¯ values calis 1, so it follows that R
culated this way for methane (unstable) and ethylene (stable) flames vary little (Fig. 6). These data do not exhibit the expected behavior; the negative Ma (unstable) flames do not have larger ¯ than the positive Ma (stable) flames, invalues of R dicating that the unstable flames are no more wrinkled than the stable flames. Furthermore, the PDF of N for stable flames is systematically broader than the PDF of N for unstable flames. This contradicts the findings of workers such as Wu et al. [4] who observed more wrinkling for negative Ma flames than positive Ma ones. The expectation that negative Ma flames will exhibit more wrinkling than positive Ma flames is based on the idea of an “isolated” flamelet element on the flamelet surface behaving only according to its local value of stretch. Clearly though, the behavior of one flamelet element is influenced by the behavior of its neighboring elements as well as the unsteady velocity field with which it is interacting. Gouldin and Miles [17] reported estimates of normalized burning rate integral data in which, for similar values of u/SL, ethylene flames have larger values of the normalized burning rate integral than do methane flames, also contrary to expectation. Additional measurements are warranted to determine if the observed behavior is general or associated with the limited range of conditions of the present data. Conclusions In conclusion, the degree of wrinkling, as mea¯ of the flames studied with positive and sured by R, negative Ma was equivalent within experimental uncertainty at similar values of u/SoL . The distribution of flamelet normals in ethylene/air V-flames was measured for the first time, and the same form of the joint PDF of N that applied for methane/air Vflames and SI engine flames applied to these flames. f values for ethylene flames were larger than those for methane flames, and measured values of f for methane flames were consistent with previous measurements. N具c典 was measured for the first time in Vflames and was found to be aligned with 具N典. Further measurements are required to determine if the unexpected wrinkling behavior for stable and unstable flames is general. Acknowledgments The authors acknowledge Sandra Sattler for her assistance in performing this research and the support of the Army Research Office, agreement number DAAD 19-991-0324. REFERENCES 1. Bray, K. N. C., and Peters, N., Turbulent Reacting Flows (P. A. Libby and F. A. Williams, eds.), Academic Press, San Diego, 1993, pp. 63–113.
FLAMELET ORIENTATIONS IN DIFFERENT MARKSTEIN-NUMBER FLAMES 2. Markstein, G. H., Non-Steady Flame Propagation, Pergamon, New York, 1964, p. 22. 3. Tseng, L.-K., Ismael, M. A., and Faeth, G. M., Combust. Flame 95:410–426 (1993). 4. Wu, M. S., Driscoll, J. F., and Faeth, G. M., Combust. Sci. Technol. 78:69–96 (1991). 5. Bradley, D., Proc. Combust. Inst. 24:247–262 (1992). 6. Haworth, D. C., and Poinsot, T. J., J. Fluid Mech. 244:405–436 (1992). 7. Echekki, T., and Chen, J. H., Combust. Flame 106:184–202 (1996). 8. Bingham, D. C., Gouldin, F. C., and Knaus, D. A., Proc. Combust. Inst. 27:77–84 (1998). 9. Knaus, D. A., Gouldin, F. C., and Bingham, D. C., unpublished work. 10. Bingham, D. C., “Crossed-Plane Laser Tomography: Direct Measurement of Flamelet Orientations and Mean Flamelet Surface Density,” M.S. thesis, Cornell University, Ithaca, NY, 1998.
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11. Knaus, D. A., Gouldin, F. C., Hinze, P. C., and Miles, P. C., SAE paper 1999-01-3543. 12. Gouldin, F. C., in Physical and Chemical Aspects of Combustion: A Tribute to Irvin Glassman, Gordon and Breach, Amsterdam, 1997, p. 433. 13. Miles, P. C., “Conditional Velocity Statistics and TimeResolved Flamelet Statistics in Premixed Turbulent VShaped Flames,” Ph.D. thesis, Cornell University, Ithaca, NY, 1991. 14. Law, C. K., in Reduced Kinetic Mechanisms for Applications in Combustion Systems (N. Peters and B. Rogg, eds.), Springer-Verlag, New York, 1993, pp. 15– 28. 15. Aung, K. T., Tseng, L.-K., Ismail, M. A., and Faeth, G. M., Combust. Flame 102:526–530 (1993). 16. Castleman, K. R., Oppenheim, A. ed Digital Image Processing, 2nd ed., Prentice Hall, Englewood Cliffs, NJ, 1996. 17. Gouldin, F. C., and Miles, P. C., Combust. Flame 100:202–210 (1995).
COMMENTS ¨ . L. Gu¨lder, National Research Council Canada, Dr. O Canada. Your mean surface density results seem to be insensitive to the normalized turbulence intensity. Could you please comment on this unexpected result? Author’s Reply. Data describing the global reaction rate of premixed turbulent flames in the form of turbulent burning velocity, uT or burning rate integral, BT increase with u/S⬚L. BT data found in [1,2] suggest an increasing trend with u/S⬚L, however, the increase is more gradual than most uT data found in the literature. Assuming the mean stretch factor, Io is unity and that the distribution of flamelet orientations (N) is independent of c, the normalized burning rate integral can be written as:
冬 冭
BT 1 qrS⬚L |rg|
Nc.
wrinkling of positive and negative Ma flames over a range of u/S⬚L. For this purpose, we chose to quantitatively describe the flamelet wrinkling with an averaged quantity, equation 5. Once again assuming that the PDF of N is independent of c, can be written as 兺¯
冬|r |冭 1
g c
Nc lb
lb data for the ethylene-air flames show an increasing trend with u/S⬚L over the range studied, off-setting the increasing trend of the mean inverse direction cosine term, and giving the unexpected result that 兺¯ shows no trend with u/S⬚L, Table 1. For larger u/S⬚L, Nc will be greater than one, and it is expected that 兺¯ will increase with u/S⬚L. Measurements are planned over a wider range of u/S⬚L to further explore Ma effects, as well as to learn more about BT and lb variation with u/S⬚L.
c
For the flames studied, Nc 1, and therefore BT depends on the mean inverse direction cosine term only. As can be seen in Table 1 for our case, the mean inverse direction cosine term, and hence BT increases gradually with u/S⬚L. The purpose of the current work was to compare the
REFERENCES 1. Ghenai, C., Gouldin, F. C., and Gokalp, I., Proc. Comb. Inst. 27:979–987 (1998). 2. Bingham, D. C., Gouldin, F. C., and Knaus, D. K., Proc. Comb. Inst. 27:77–84 (1998).
MEASUREMENTS OF FLAMELET ORIENTATIONS IN PREMIXED FLAMES WITH POSITIVE AND NEGATIVE MARKSTEIN NUMBERS D. A. KNAUS and F. C. GOULDIN Sibley School of Mechanical and Aerospace Engineering Cornell University Ithaca, NY 14853, USA
Measurements of instantaneous flamelet surface orientation of preferential diffusion stable, ethylene/ air and unstable, methane/air flames are presented for three different values of normalized turbulence intensity, u/SoL , where SoL is the unstretched laminar flame speed. Instantaneous flamelet orientations were measured in three dimensions via crossed-plane imaging. Repeated measurements were used to estimate probability density functions (PDFs) of flamelet orientation. A one parameter fit of the PDF of flamelet orientations was applied that has been applied previously to data from V-flames and from spark ignition engine flames. The fits of the PDF were in good agreement with measurements. The fit parameter, f, which describes the breadth of the distribution of flamelet orientations, was found to be systematically larger for stable flames than for unstable flames at similar values of u/SoL . The degree of wrinkling of the ¯ averaged across the flame brush, and flamelet isdescribed quantitatively by the flamelet surface density, R, did not vary for stable and unstable flames, contrary to expectation. f data from the current work are in good agreement with data from previous experiments. A 具c典 constant contour image is presented to give an indication of the shape and width of the turbulent flame brush. Normals to 具c典 constant surfaces, N具c典, were measured for the first time in V-flames and were found to be colinear with the mean of the flamelet normals, 具N典.
For steady, laminar flames with small Karlovitz number, Ka, the effect of stretch on the flame speed can be described by the equation [2]
Introduction Premixed turbulent flames in the flamelet regime are characterized by regions of reactants and regions of products separated by a thin flame sheet, or flamelet. The primary assumption of the flamelet description is that a surface element on the flamelet has the structure of a perturbed laminar flame and is propagating locally with respect to the reactants at the corresponding laminar burning velocity modified to account for the local perturbation of the flamelet element. The burning intensity is related to the degree of wrinkling of the flamelet, which increases the flamelet surface area per unit volume and therefore increases the rate of reactant consumption per unit volume. For the flamelet regime, the mean rate of creation of products, 具w典, can be written 具w典 qrSLo IoR reactant density, SoL
(1)
where qr is the is the unstretched laminar flame speed, Io is a factor that accounts in the mean for perturbation effects on SoL , and R is the mean flamelet surface area to volume ratio, or the flamelet surface density [1]. When written this way, information about the degree of wrinkling of the flamelet is contained in R, and information about the effect of the wrinkling on the local flame speed is contained in Io.
SL [1 MaKa]1 SoL
(2)
In equation 2, SL is the flame speed and Ma is the Markstein number. Ka is a dimensionless number quantifying the stretch magnitude, and Ma is a dimensionless number containing the Markstein Length, L. The value of L and in turn Ma determines the effect on the flame speed of the flamelet stretch. For positive Ma flames, the flame speed decreases with positive stretch, and for negative Ma flames, the flame speed increases with positive stretch. For a positive Ma flame, a surface element with positive curvature, such as that at the tip of a finger of products protruding into a reactant region, is slowed down, and a surface element with negative curvature, such as that at the tip of a finger of reactants protruding into a product region, speeds up. This is referred to as a stable preferential diffusion condition, because a perturbation of the flamelet surface (i.e., a wrinkle) is damped [3]. An unstable preferential diffusion condition exists for negative Ma flames. A surface element with positive curvature, such as that located at the tip of a finger of products protruding into a reactant region, speeds
367
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TURBULENT COMBUSTION—Premixed Combustion Measurements
up, and a surface element with negative curvature, such as that located at the tip of a finger of reactants protruding into a product region, slows down. The result is an amplification of the wrinkling of the flamelet. Positive Ma flames are preferential diffusion stable, and negative Ma flames are preferential diffusion unstable [3]. This effect is referred to as preferential diffusion because the value of Ma for a flame is dependent on, among other things, the relative mass diffusivities of the fuel and oxidizer. For flames where the fuel and oxidizer have significantly different diffusivities, fuel and oxidizer approach the flame zone at different rates due to stretch, which affects the flame speed [3]. Ma describes a flame’s response to stretch and is therefore dependent on the relative diffusivities of fuel and oxidizer. Numerous workers have provided evidence of the importance of Ma and Le, Lewis number, effects on the surface topography, and burning rate of premixed flames. Wu et al. [4] presented experimental results indicating variations in flamelet wrinkling for different Ma turbulent flames. Bradley [5] presented data that indicate the need to correlate experimental values of turbulent burning velocity with KaLe. He also stated that it is ultimately more logical to correlate with KaMa, pending the availability of Ma data, which are becoming more and more available. Haworth and Poinsot [6] demonstrated the importance of Le effects in premixed turbulent flames using numerical calculations, as did Echekki and Chen [7]. The primary purpose of the current work was to observe and quantify the degree of wrinkling of positive and negative Ma flames. Two flames were studied, one with a positive Ma, and one with a negative Ma, at various values of u/SoL, where u is the turbulence intensity in the reactants approaching the flame. Flamelet orientation measurements were made via crossed-plane imaging [8–11]. Crossedplane imaging allowed the instantaneous flamelet surface normal vector, N, to be measured in three dimensions via two simultaneous laser tomography measurements made in orthogonal planes. The two tomography laser sheets intersected along a line referred to as the measurement line. At points where the measurement line intersected the flamelet the orientation of the flamelet surface within the plane of each laser sheet was measured. Because the orientations of the laser sheets in space were known, N could be measured in three dimensions. N was defined as a unit vector normal to the flamelet surface oriented in the direction of flamelet propagation, that is, toward reactants. The probability density function (PDF) of N was estimated from repeated measurements and used to help quantitatively describe the degree of wrinkling of the flamelet. It was expected that a positive Ma number (stable) flame would exhibit a lower degree of wrinkling than a
negative Ma (unstable) flame for the same value of u/SoL. Specifically, N data were obtained for three stable ethylene/air flames and three unstable methane/air flames. These data were used to test a single parameter PDF fit that was previously applied to V-flame [8] and spark ignition engine (SI) flame data [11] and is of the form P(,h)sin ddh A|N • ny|exp (2/f2) sin ddh
(3)
Here N is expressed in terms of spherical coordinates and h, and A is a normalization constant. The vector dot product term accounts for crossingweighting in the measured PDFs. Crossed-plane imaging measurements are inherently crossingweighted: the probability associated with a flamelet of a certain orientation being measured is weighted by the probability that a flamelet having that orientation crosses the measurement line [8]. ny is a unit vector aligned with the measurement line. The following expression relates N data to R[12]: R
冬|r |冭 1
g
nc
(4)
c
The bracket denotes an ensemble averaged quantity. rg is the direction cosine of N and a line g, which is defined as spanning the turbulent flame brush normal to mean progress variable, 具c典, constant surfaces. nc is the flamelet crossing density, the average number of flamelet crossings of g per unit length. The c subscript indicates that the average is crossingweighted. The apparent singularity in equation 4 for rg 0 is not realized due to the crossing-weighting of the average, since a surface parallel to a line does not cross the line [8,9]. R describes the average degree of wrinkling of the flamelet at a point in space. The average degree of wrinkling across g is quantified by the expression ¢ 1 R lb
⬁
冮
⬁
Rdg
(5)
where lb is the flame brush thickness. The instantaneous progress variable, c, describes the progress of reaction of premixed flames. It is defined to be 0 in reactants and 1 in products. For flames in the flamelet regime, the reaction zone can be approximated as infinitesimally thin, and the instantaneous c field is bivalued, 0 in reactants and 1 in products. The mean progress variable, 具c典, field describes the shape and width of the turbulent flame brush. The 具c典 field can be measured within the planes of the two laser sheets used for crossed-plane imaging. Images are made binary to obtain instantaneous c fields and then averaged. The flame brush thickness, lb, is defined as
FLAMELET ORIENTATIONS IN DIFFERENT MARKSTEIN-NUMBER FLAMES
369
TABLE 1 Flame conditions and summary of results Flame Methane 1 Methane 2 Methane 3 Ethylene 1 Ethylene 2 Ethylene 3
U
SoL (cm/s)
u/SoL
Ma
f (⬚)
具1/|rg|典c
lb (mm)
¯ (mm1) R
0.65 0.65 0.65 0.61 0.61 0.61
15 15 15 26 26 26
0.60 0.90 1.16 0.51 0.77 1.00
0.14 0.14 0.14 2.77 2.77 2.77
28 30 30 30 35 36
1.12 1.14 1.14 1.14 1.20 1.21
0.93 0.81 0.78 0.94 0.91 1.11
1.20 1.41 1.46 1.22 1.32 1.09
Fig. 1. SoL and Ma as a function of equivalence ratio for methane-air (closed symbols) and ethylene-air (open) flames. The SoL curves are fits of the data found in Law [14]. The Ma curves are based on correlations found in Aung et al. [15].
lb
⬁
冮
⬁
具c典 (1具c典) dg
(6)
and can be evaluated from 具c典 field image data by integration. From crossed-plane 具c典 field images, normals to 具c典 constant surfaces, N具c典, can be measured along the measurement line in a way analogous to that for finding N. In previous crossed-plane imaging measurements of turbulent methane/air flames, it was found that the surface-weighted PDF of N is uniform in the azimuthal angle when written in spherical coordinates with the polar axis aligned with 具N典 [8], while N具c典 and 具N典 were found to be colinear for the SI engine flames studied in Ref. [11]. A secondary purpose of the current work was to determine whether 具N典 and N具c典 are colinear for V-flames. Experiment Experiments were performed on the V-flame burner used in several previous studies in this laboratory [8–10,13,14]. Fuel and air flow rates were measured with mass flow meters. A portion of the air flow was diverted through a blast-atomizer-type
seed particle generator, within which it was seeded with silicone oil droplets, and then passed through a cyclone separator. Droplet sizes were estimated to be no more than a few micrometers [13]. The reactants were mixed in a plenum at the base of the burner. A square mesh wire grid with approximately 4.5 squares/cm2 and a 1 mm wire diameter were positioned 50 mm upstream of the burner exit. Measurements were made at an axial distance of 40 mm downstream of the stabilizer rod, which was located at the burner exit. The flames studied are described in Table 1; they were chosen so that measurements could be made on both positive and negative Ma flames over similar ranges of u/SoL. Lean flames are preferred to avoid soot formation, which can interfere with laser tomography measurements. Values of SoL are taken from Ref. [14]. Linear correlations relating the Ma to the equivalence ratio, U, for methane/air and ethylene/air flames found in Aung et al. [15] were used to estimate Ma. For sufficiently lean mixtures, ethylene/air flames have positive Ma values and methane/air flames have negative Ma values, as can be seen in the plots of Ma and SoL in Fig. 1. For each fuel studied, a fixed value of U was chosen so that for each fuel Ma is fixed. u was then varied, by varying the flow rate, so that flames with similar values of u/SoL were measured for each fuel. u was determined from a linear fit of u to the area averaged mean jet velocity based on previous laser doppler velocimetry measurements performed on the burner [8]. Values of Ka for the flames studied were estimated to be on the order of 106 and were small enough that equation 2 was expected to apply to these flames [2]. The frequency-doubled 532 nm beam from a Nd:YAG pulsed laser (7 ns pulse duration) was used for tomographic visualization. The 6.4 mm diameter beam was focused and split into two mutually orthogonal light sheets approximately 70 mm wide, with waist thicknesses of less than 0.5 mm. The polarization of each beam was rotated via half-wave plates so that it laid in the plane of its light sheet. The two sheets were positioned so that each crossed over the burner at a 45⬚ angle to the vertical, with
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TURBULENT COMBUSTION—Premixed Combustion Measurements
Fig. 2. Schematic representation of crossed-plane imaging measurements on a turbulent V-flame. The light from orthogonal laser sheets is scattered by oil droplets seeded in the reactants.
Fig. 3. 具c典 regions for flame Methane 1. Also shown is the position of the measurement line. The 具c典 domain, from 0 to 1, is divided into tenths, each represented by a grayscale color. The boundary of color regions represent the location of 具c典 constant contours.
their line of intersection perpendicular to the stabilizing rod. Fig. 2 shows schematically the physical appearance of the laser sheets interacting with the V-flame and the laboratory coordinate system with the y axis aligned with the measurement line, the z axis vertical, and the x axis aligned with the stabilizer rod such that the coordinate system is right-handed. Tomographic images were recorded simultaneously with two digital cameras, each having 256 color grayscale, 652 494 pixel CCD arrays. The cameras were positioned perpendicular to the laser sheets; that is, one sheet was parallel to the camera image plane, and the second was edge-on for each camera. They were fitted with 50 mm focal length lenses and 10 mm extension tubes, which provided a field of view of approximately 18 mm 24 mm. The camera resolution was greater than 0.2 mm. Polarizing filters mounted on the end of the lenses
blocked scattered light from the edge-on sheet. Hence, each sheet was imaged individually, with negligible interference from the light scattered from the other [8]. Ten nanometer bandwidth, laser line filters were used to reduce the amount of flame radiation reaching the cameras. The external trigger of the laser was used to trigger the cameras, and the camera exposure times were set to the minimum value, 10 ms. Oil droplets scattered the laser light in the reactants but were vaporized by the flame. The images were therefore bright in the reactants and dark in the products; the boundary between the two regions represents the curve of the flamelet surface within the plane of the laser sheet, the flamelet boundary. Images were recorded and saved for postprocessing. Image processing consisted of a thresholding step, which made the images binary, followed by the binary image operations of opening and closing [16] that filtered island features of diameters less than the opening/closing diameter, which was set at 5 pixels or ⬃0.2 mm. Flamelet crossings were identified in each image as points where the flamelet boundary intersected the measurement line. The flamelet boundary was found by an edge-finding algorithm and then fitted over a fit-width of 100 pixels at each crossing location. From a fit of the flamelet, its slope was determined at flamelet crossing points and used to define tangent vectors to the flamelet boundary. The tangent vectors in orthogonal image pairs for a given flamelet crossing were orthogonal to each other and tangent to the flamelet surface. The normalized cross-product of these tangent vectors was N. Further details on image processing can be found in Refs. [8] and [10]. For each flame studied, 1000 crossed-plane image pairs were recorded. The 具c典 fields in the planes of the two laser sheets were measured from the recorded crossed-plane images by averaging the images after they were made binary. Fig. 3 shows different 具c典 regions for flame Methane 1. The 具c典 values are grouped in tenths, each represented by a grayscale color. The boundaries between grayscale regions in the image represent 具c典 constant contours, the intersections of 具c典 surfaces with the plane of the laser sheet. The normals to 具c典 constant surfaces can be measured the same way N is measured: tangent vectors to a 具c典 constant contour are calculated from a fit of the 具c典 constant contour for both 具c典 field images, and the normalized cross-product of the tangent vectors is N具c典.
Results and Discussion For turbulent V-flames, it is expected that 具c典 surfaces are planes, and in the y-z (vertical) plane, it is expected that 具c典 surfaces appear as straight rays
FLAMELET ORIENTATIONS IN DIFFERENT MARKSTEIN-NUMBER FLAMES
371
Fig. 4. Projections onto the x-z plane of the tips of N具c典, 具c典0.5, and of 具N典 vectors for the flames studied.
originating from the stabilizer rod. This expected behavior can be observed qualitatively in Fig. 3, although the 具c典 constant contours are not perfectly straight. Note that the image plane shown in Fig. 3 is not coincident with the y-z plane but is 45⬚ from the vertical. Fig. 4 shows the projections of the end points of N具c典, 具c典 0.5, onto the x-z plane for all the flames studied. It is expected that N具c典 will have a small negative z component (down in the laboratory coordinate system) due to the shape of a V-flame and no x component. In Fig. 4, the data for N具c典 all have a small positive x component. Values of the angle that the projection of N具c典 onto the x-z plane make with the z axis are in the range of 6–16⬚. N具c典 vectors not being in the vertical plane is attributable to two possible causes. First, there is uncertainty in the orientation of the stabilizing rod with respect to the measurement line, which is estimated to be 5⬚. Second, there may be some non-uniformity in the mean axial velocity profiles exiting the burner. Although N具c典 are deviating slightly from the expected orientation, the important conclusion is that N具c典 are aligned with 具N典 to within a small difference, also shown in Fig. 4. To present the N data, we defined a spherical coordinate system with a polar angle, ; an azimuthal angle, h; and its polar axis aligned with 具N典. Because 具N典 is aligned with N具c典, either vector could be used to define the pole of the spherical coordinate system. Instantaneous flamelet normals were measured over a range of 具c典 along the measurement line. Due to the limited number of flamelet normals measured, it was not possible to obtain statistically valid estimates of N data PDFs for narrow 具c典 ranges. Instead, all measured normals for a flame condition were
Fig. 5. Measured marginal PDFs in (top) and h (bottom) for flame Ethylene 2. Also shown are fits based on a fit of the joint PDF of the form P(,h)A[N•ny]exp(2/ f2). For the flame shown, f35⬚.
grouped into one distribution. For SI engine flames, the PDFs of flamelet normals were found to vary only slightly over the range 0.1 ⬍ 具c典 ⬍ 0.75 [11]. Intuitively, one would expect that the distribution of flamelet orientations will broaden in the extreme 具c典 regions, 具c典 → 0 or 1. This expected broadening of N is expected to have a only a small effect on the crossing-weighted PDF of N and on R, since for 具c典 → 0 or 1, nc is small; that is, relatively few normals from the flame brush edges are measured and therefore contribute to the PDF estimate or to R calculations. Distributions of flamelet orientations can be presented in the form of marginal PDFs in and h, P() and P(h), which are shown in Fig. 5a and b for the ethylene 2 flame. P() is formed by generating a histogram of measured values, independent of h, and normalizing. P(h) is generated by forming a histogram of h values independent of and normalizing. Also shown in Fig. 5a and b are marginal PDFs based on a fit of the joint PDF, equation 3. The fit and data are in good agreement for ethylene 2 and for all of the flames studied. This demonstrates that the form of the PDF fit applies to ethylene flames as well as methane flames. Much of the scatter in the PDF data can be attributed to uncertainty associated with the number of samples in each histogram bin. An error bar is shown in Fig. 5a and b
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TURBULENT COMBUSTION—Premixed Combustion Measurements
Fig. 6. PDF fit parameter, f, and average flamelet sur¯ plotted as a function face density across the flame brush, R, of normalized turbulence intensity. Also shown are previous f values for methane-air flames [8] measured at the same axial distance from the V-flame stabilizing rod as the current work. The lines represent linear fits of f values for the current ethylene flames and for both the current and previous methane flames combined.
for uncertainty due to bin sample size. Removing the crossing-weighting term, |N•ny|, from equation 3 gives the corresponding fit of the surface-weighted [8,9] joint PDF. When the spherical coordinate system is defined with the pole aligned with 具N典, or equivalently N具c典, the surface-weighted joint PDF is independent of h, and the distribution of N is described by the surface-weighted marginal PDF in alone, which is quantified in turn by the fit parameter, f. Flames with narrow distributions of N have narrow, peaked marginal PDFs in , and small values of f. Conversely, flames with broad distributions of N have broad marginal PDFs in , and large values of f. Values of f for the flames studied are given in Table 1 and plotted as a function of u/SoL in Fig. 6. Also presented in Fig. 6 are data for methane/air flames from previous measurements [8]. Inspection of the linear fits of the data show that f values for the ethylene flames are systematically higher than those for the methane flames, the opposite of the expected behavior. Comparison of the current data and the previous data give an indication of the repeatability of the experiment. The degree of wrinkling of the flames studied is ¯ and can be determined from equaquantified by R tion 5. A single value of f is assumed for all g, so that the mean inverse direction cosine term can be factored out of the integral. Convolution of the fit of the joint PDF with the inverse direction cosine term in equation 4 yields the mean inverse direction cosine term for a given f value (Table 1). Also shown in Table 1 are values of the flame brush thickness, lb. For the flames studied, the integral of nc across g, or the average number of flamelet crossings of g, ¯ 具1/|rg|典c/lb. R ¯ values calis 1, so it follows that R
culated this way for methane (unstable) and ethylene (stable) flames vary little (Fig. 6). These data do not exhibit the expected behavior; the negative Ma (unstable) flames do not have larger ¯ than the positive Ma (stable) flames, invalues of R dicating that the unstable flames are no more wrinkled than the stable flames. Furthermore, the PDF of N for stable flames is systematically broader than the PDF of N for unstable flames. This contradicts the findings of workers such as Wu et al. [4] who observed more wrinkling for negative Ma flames than positive Ma ones. The expectation that negative Ma flames will exhibit more wrinkling than positive Ma flames is based on the idea of an “isolated” flamelet element on the flamelet surface behaving only according to its local value of stretch. Clearly though, the behavior of one flamelet element is influenced by the behavior of its neighboring elements as well as the unsteady velocity field with which it is interacting. Gouldin and Miles [17] reported estimates of normalized burning rate integral data in which, for similar values of u/SL, ethylene flames have larger values of the normalized burning rate integral than do methane flames, also contrary to expectation. Additional measurements are warranted to determine if the observed behavior is general or associated with the limited range of conditions of the present data. Conclusions In conclusion, the degree of wrinkling, as mea¯ of the flames studied with positive and sured by R, negative Ma was equivalent within experimental uncertainty at similar values of u/SoL . The distribution of flamelet normals in ethylene/air V-flames was measured for the first time, and the same form of the joint PDF of N that applied for methane/air Vflames and SI engine flames applied to these flames. f values for ethylene flames were larger than those for methane flames, and measured values of f for methane flames were consistent with previous measurements. N具c典 was measured for the first time in Vflames and was found to be aligned with 具N典. Further measurements are required to determine if the unexpected wrinkling behavior for stable and unstable flames is general. Acknowledgments The authors acknowledge Sandra Sattler for her assistance in performing this research and the support of the Army Research Office, agreement number DAAD 19-991-0324. REFERENCES 1. Bray, K. N. C., and Peters, N., Turbulent Reacting Flows (P. A. Libby and F. A. Williams, eds.), Academic Press, San Diego, 1993, pp. 63–113.
FLAMELET ORIENTATIONS IN DIFFERENT MARKSTEIN-NUMBER FLAMES 2. Markstein, G. H., Non-Steady Flame Propagation, Pergamon, New York, 1964, p. 22. 3. Tseng, L.-K., Ismael, M. A., and Faeth, G. M., Combust. Flame 95:410–426 (1993). 4. Wu, M. S., Driscoll, J. F., and Faeth, G. M., Combust. Sci. Technol. 78:69–96 (1991). 5. Bradley, D., Proc. Combust. Inst. 24:247–262 (1992). 6. Haworth, D. C., and Poinsot, T. J., J. Fluid Mech. 244:405–436 (1992). 7. Echekki, T., and Chen, J. H., Combust. Flame 106:184–202 (1996). 8. Bingham, D. C., Gouldin, F. C., and Knaus, D. A., Proc. Combust. Inst. 27:77–84 (1998). 9. Knaus, D. A., Gouldin, F. C., and Bingham, D. C., unpublished work. 10. Bingham, D. C., “Crossed-Plane Laser Tomography: Direct Measurement of Flamelet Orientations and Mean Flamelet Surface Density,” M.S. thesis, Cornell University, Ithaca, NY, 1998.
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11. Knaus, D. A., Gouldin, F. C., Hinze, P. C., and Miles, P. C., SAE paper 1999-01-3543. 12. Gouldin, F. C., in Physical and Chemical Aspects of Combustion: A Tribute to Irvin Glassman, Gordon and Breach, Amsterdam, 1997, p. 433. 13. Miles, P. C., “Conditional Velocity Statistics and TimeResolved Flamelet Statistics in Premixed Turbulent VShaped Flames,” Ph.D. thesis, Cornell University, Ithaca, NY, 1991. 14. Law, C. K., in Reduced Kinetic Mechanisms for Applications in Combustion Systems (N. Peters and B. Rogg, eds.), Springer-Verlag, New York, 1993, pp. 15– 28. 15. Aung, K. T., Tseng, L.-K., Ismail, M. A., and Faeth, G. M., Combust. Flame 102:526–530 (1993). 16. Castleman, K. R., Oppenheim, A. ed Digital Image Processing, 2nd ed., Prentice Hall, Englewood Cliffs, NJ, 1996. 17. Gouldin, F. C., and Miles, P. C., Combust. Flame 100:202–210 (1995).
COMMENTS ¨ . L. Gu¨lder, National Research Council Canada, Dr. O Canada. Your mean surface density results seem to be insensitive to the normalized turbulence intensity. Could you please comment on this unexpected result? Author’s Reply. Data describing the global reaction rate of premixed turbulent flames in the form of turbulent burning velocity, uT or burning rate integral, BT increase with u/S⬚L. BT data found in [1,2] suggest an increasing trend with u/S⬚L, however, the increase is more gradual than most uT data found in the literature. Assuming the mean stretch factor, Io is unity and that the distribution of flamelet orientations (N) is independent of c, the normalized burning rate integral can be written as:
冬 冭
BT 1 qrS⬚L |rg|
Nc.
wrinkling of positive and negative Ma flames over a range of u/S⬚L. For this purpose, we chose to quantitatively describe the flamelet wrinkling with an averaged quantity, equation 5. Once again assuming that the PDF of N is independent of c, can be written as 兺¯
冬|r |冭 1
g c
Nc lb
lb data for the ethylene-air flames show an increasing trend with u/S⬚L over the range studied, off-setting the increasing trend of the mean inverse direction cosine term, and giving the unexpected result that 兺¯ shows no trend with u/S⬚L, Table 1. For larger u/S⬚L, Nc will be greater than one, and it is expected that 兺¯ will increase with u/S⬚L. Measurements are planned over a wider range of u/S⬚L to further explore Ma effects, as well as to learn more about BT and lb variation with u/S⬚L.
c
For the flames studied, Nc 1, and therefore BT depends on the mean inverse direction cosine term only. As can be seen in Table 1 for our case, the mean inverse direction cosine term, and hence BT increases gradually with u/S⬚L. The purpose of the current work was to compare the
REFERENCES 1. Ghenai, C., Gouldin, F. C., and Gokalp, I., Proc. Comb. Inst. 27:979–987 (1998). 2. Bingham, D. C., Gouldin, F. C., and Knaus, D. K., Proc. Comb. Inst. 27:77–84 (1998).