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High-repetition rate measurements of temperature and thermal dissipation in a non-premixed turbulent jet flame
Proceedings of the
Combustion Institute
Proceedings of the Combustion Institute 30 (2005) 691–699
www.elsevier.com/locate/proci
High-repetition rate measurements of temperature and thermal dissipation in a non-premixed turbulent jet flame G.H. Wang, N.T. Clemens*, P.L. Varghese Department of Aerospace Engineering and Engineering Mechanics, Center for Aeromechanics Research, The University of Texas at Austin, Austin, TX 78712-1085, USA
Abstract High-repetition rate laser Rayleigh scattering is used to study the temperature fluctuations, power spectra, gradients, and thermal dissipation rate characteristics of a non-premixed turbulent jet flame at a Reynolds number of 15,200. The radial temperature gradient is measured by a two-point technique, whereas the axial gradient is measured from the temperature time-series combined with Taylors hypothesis. The temperature power spectra along the jet centerline exhibit only a small inertial subrange, probably because of the low local Reynolds number (Red 2000), although a larger inertial subrange is present in the spectra at off-centerline locations. Scaling the frequency by the estimated Batchelor frequency improves the collapse of the dissipation region of the spectra, but this collapse is not as good as is obtained in non-reacting jets. Probability density functions of the thermal dissipation are shown to deviate from lognormal in the low-dissipation portion of the distribution when only one component of the gradient is used. In contrast, nearly log-normal distributions are obtained along the centerline when both axial and radial components are included, even for locations where the axial gradient is not resolved. The thermal dissipation PDFs measured off the centerline deviate from log-normal owing to large-scale intermittency. At one-half the visible flame length, the radial profile of the mean thermal dissipation exhibits a peak off the centerline, whereas farther downstream the peak dissipation occurs on the centerline. The mean thermal dissipation on centerline is observed to increase linearly with downstream distance, reach a peak at the location of maximum mean centerline temperature, and then decrease for farther downstream locations. Many of these observed trends are not consistent with equivalent non-reacting turbulent jet measurements, and thus indicate the importance of understanding how heat release modifies the turbulence structure of jet flames. 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Turbulent flame; Scalar dissipation; Rayleigh scattering
1. Introduction Detailed measurements of mean and fluctuating scalars, such as species mass fractions and *
Corresponding author. Fax: +1 512 471 3788. E-mail address: [email protected] (N.T. Clemens).
temperature, have been critical to developing an improved understanding of the physics of turbulent non-premixed flames [1–4]. Of particular importance are the mixture fraction n and its gradients, because in the flamelet theory the flame structure is fundamentally related to the value of n at stoichiometric conditions and the rate of scalar dissipation, v ” 2D (,n Æ ,n), where D is the
1540-7489/$ - see front matter 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2004.08.269
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diffusivity. The scalar dissipation rate, which is a measure of the mixing rate, limits the reaction rate under mixing limited conditions, and affects the degree of non-equilibrium under finite chemistry conditions. Because of its importance to combustion, measurements of v in turbulent non-premixed flames have received a lot of attention in recent years [5–7], but detailed statistical measurements of the type that exist for non-reacting flows are still relatively sparse. It can be argued that temperature does not play as fundamental a role as mixture fraction in determining the flame characteristics, but its fluctuations and gradients do provide important information about the underlying mixture fraction structure. This can be seen by considering the thermal dissipation rate vT = 2a (,T Æ ,T), where T is the temperature and a is the thermal diffusivity, which is related to the rate of thermal mixing, or alternatively to the rate at which thermal inhomogeneities are removed by diffusion. Importantly, under the assumption of the state relationship T = T (n) and unity Lewis number, the scalar and thermal dissipation rates are related as vT = v (dT/dn)2 [8]. As will be discussed in Section 4, in some regions of the flame, dT/dn is approximately constant, and so the thermal dissipation rate is proportional to the scalar dissipation rate. Thermal mixing is also important because it affects high-temperature chemical reaction processes and can be important in the development and validation of turbulent flame models [8]. For these reasons, fluctuating temperature and thermal dissipation rates have been measured in a number of studies by using, e.g., dual-thermocouple measurements [9,10], two-point laser Rayleigh measurements [11,12], and planar laser Rayleigh imaging [8]. An important issue with such measurements is that the requirement to obtain fully spatially and temporally resolved measurements of the finest scales of turbulence is very stringent and this makes dissipation measurements, in particular, challenging [13]. It is clear from a careful study of the literature that the Batchelor scale is rarely resolved, even in turbulent flame studies that explicitly seek to measure the dissipation rate. The objective in this study was to make highquality, high-repetition rate (10 kHz), two-point laser Rayleigh temperature measurements in a weakly co-flowing turbulent non-premixed jet flame at a Reynolds number of 15,200, with high signal-to-noise ratio (50 in room air) and where the finest scales of turbulence are spatially and temporally resolved. These two-point temperature data were used to obtain temperature power spectra and detailed statistics of the thermal dissipation rate. The flame studied here is similar to the TNF simple jet flame (DLR_A), which is used as a benchmark flame for the TNF Workshop [14–17].
2. Experimental setup The flow studied was a weakly co-flowing jet flame. The co-flow air was filtered to remove particles larger than 0.2 lm and then passed through a flow conditioning section. The co-flow velocity was 0.45 m/s. The fuel issued from a long tube with inside diameter d = 7.75 mm. The test section had a 0.75 m · 0.75 m cross-section. The whole jet flow facility was mounted on a traverse that was driven by stepper motors to provide positioning in the radial and axial directions. The fuel composition used in this study was 22.1% CH4, 33.2% H2, and 44.7% N2 (by volume), which gives a stoichiometric mixture fraction of 0.167. The fuel gases were metered by pressure regulators and monitored by mass flowmeters to an accuracy of ±1.0 SLPM for N2 and ±0.5 SLPM for CH4 and H2. The Rayleigh cross-section of this fuel has been shown to vary by ±3% across the whole flame [15]. The source Reynolds number was Red = U0d/m0 = 15,200 (where m0 is the kinematic viscosity of the fuel and U0 is the jet exit bulk velocity), and the measurements were taken at downstream locations from x/d = 40 to 80. Here, x and r are the axial and radial coordinates, respectively. The visible flame length was at about x/d = 84, and the stoichiometric flame length, estimated based on data in the TNF database, was at about x/d = 60. The laser Rayleigh system is based on a diodepumped Nd:YAG laser operated at 71 W average power at 532 nm and with a 10 kHz repetition rate. The laser beam was focused into the test section by using a 300 mm focal length lens. The beam diameter was measured to be about 0.3 mm. An external photodiode was used to correct for variations in the laser pulse energy on a shot-by-shot basis. Rayleigh scattered light was collected using custom-designed optics that consisted of a pair of 150 mm diameter plano-convex lenses, one 50.8 mm diameter meniscus lens and one 50.8 mm diameter double-convex lens. The lens system was designed with ZEMAX and produced an aberration-limited blur-spot of less than 34 lm. The working f# was 2.4, and the magnification was 0.685. Two-point measurements were made by imaging the scattered light onto a broadband hybrid cube beam splitter, which reflected and transmitted the split signal onto two different PMTs. Two 200 lm slits were placed in front of the PMTs to define the spatial resolution (i.e., length of the beam imaged). The slit width in the image plane corresponded to 300 lm in the object plane. These two slits were arranged such that the separation of probe volumes in the flow was 300 lm. The PMT and photodiode outputs were read by gated integrators operated with gate widths of 300 ns. The integrated signals were synchronously sampled by a 12-bit A/D converter at 10 kHz.
G.H. Wang et al. / Proceedings of the Combustion Institute 30 (2005) 691–699
For constant Rayleigh cross-section, the temperature is derived from the formula: T ¼ I R;ref T ref =I R , where I R;ref is the reference Rayleigh scattering signal from air at room temperature (Tref). The two-point instantaneous temperature signals were used to determine the instantaneous radial temperature gradient by using the approximation oT/or = DT/Dr, where DT is the temperature difference of the two measurements and Dr = 300 lm is the probe separation distance. The axial gradient is estimated by the time-series and Taylors hypothesis, oT/ ox = (1/U) DT/Dt, where U is the local velocity obtained from [17]. The error in using Taylors hypothesis is estimated to be about 10% at the jet centerline [11,18]. For convenience, we define the thermal dissipation based on single components of the gradient vector: vT ;r ¼ 2aðoT =orÞ2 , and vT ;2D ¼ vT ;r þ vT ;x , vT ;x ¼ 2aU 2 ðoT =otÞ2 , where the thermal diffusivity is computed from the formula a = 2 · 105 (T/300)1.8 (m2/s) [11]. 3. Resolution estimation The finest spatial structures in the scalar field are of the order of the Batchelor scale, which is defined as kB = g Sc1/2, where g = (m3/Æeæ)1/4 is the Kolmogorov scale, m is the kinematic viscosity, Æeæ is the mean rate of kinetic energy dissipation, and Sc = m/D is the Schmidt number. Using measurements of Æeæ in non-reacting round jets [19], the Batchelor scale can be shown to be equal to kB = 2.3 d Red3/4 Sc1/2, where Red is the local Reynolds number and is defined as UCd/m, with UC the jet centerline velocity and d the full width at half maximum of the velocity profile. In non-reacting jets, Red is approximately equal to Red but this is not true in jet flames [20]. For example, the local Reynolds numbers at the downstream stations studied are shown in Table 1. The kinematic viscosity was estimated to be that of air at the measured centerline temperature according to the formula m = m0 (TC/T0)1.7. For time-series measurements, the highest frequency present in the flow is expected to be the ‘‘convective’’ Batchelor frequency fB = U/(2pkB), where U is the local mean velocity. Note that the Batchelor frequency can be quite large in jet
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flames and is not always resolved because signalto-noise ratio considerations usually necessitate relatively low bandwidths. To estimate fB in this study, the velocities used were taken from [17]. The velocities and the estimated Batchelor scales and Batchelor frequencies are shown in Table 1 for various locations along the jet centerline. Off-centerline mean velocities were also taken from [17] and used for scaling the data where appropriate, but these values are not shown in the table. Table 1 shows that the current twopoint spatial separation of 300 lm is equal to kB at x/d = 40 and is about a factor of 2 smaller than kB farther downstream. This suggests that the radial gradient measurement is fully spatially resolved. The convective Batchelor frequency fB is about 2 times the frequency resolution at x/ d = 40, and the errors in the axial dissipation are expected to be about 10%. The sampling frequency of this study is sufficient to resolve the convective Batchelor frequency (and hence the Batchelor scale) along the centerline for x/ d P 60. The measurements were limited to stations of x/d = 40 and larger so that the resolution of the dissipation scales could be maintained, at least for the radial gradient. 4. Results and discussion Figure 1 shows the measured mean and rms radial temperature profiles at three downstream stations (x/d = 40, 60, and 80). For comparison, the temperature profiles from the TNF database [14] are also shown, and it is seen that the current measurements agree very well with the database. Temperature power spectra are shown in Fig. 2. In Fig. 2A, spectra are shown along the centerline of the jet flame at the five axial stations in the range x/d = 40–80. The frequency is normalized by the convective Batchelor frequency, and the power spectral density is normalized by T 2rms =fB . The spectra have also been corrected to remove the contribution from shot-noise by using the procedure developed in [21]. This procedure requires that the measurement cut-off frequency be higher than the Batchelor frequency so that the power of the noise fluctuations can be determined. Since this was not the case for all measure-
Table 1 Jet flame conditions and estimates of resolution requirements x/d
TC (K)
U C =U C0
d/d
Red
kB (mm)
fB (kHz)
40 50 60 70 80
1580 1745 1840 1680 1470
0.36 0.29 0.24 0.21 0.18
5.3 6.7 8.0 9.3 10.6
2090 1760 1610 1870 2350
0.32 0.45 0.58 0.60 0.58
9.3 5.3 3.4 2.8 2.5
Velocity data from [17], U C0 ¼ 51:6 m=s is the mean jet exit centerline velocity, which is different from the jet exit bulk velocity U0; temperature data from TNF database [14].
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Fig. 1. Radial profiles of (A) mean and (B) rms temperature at three different axial stations (x/d = 40, 60, and 80). Data from TNF workshop database [14] are shown for comparison.
Fig. 2. Fluctuating temperature power spectra. (A) Spectra at the jet flame centerline (corrected except at x/d = 40 and 50); (B) along ray r/d = 0.4 (corrected except at x/d = 40); and (C) at x/d = 80 and different radial locations, all corrected.
ment locations, the correction was not applied in all cases (see figure caption). All of the spectra exhibit a similar appearance in that they are relatively flat at low frequency, begin rolling off for f/fB > 102, and then roll-off more rapidly in the dissipation range (f/fB > 0.4). An inertial subrange, where the power scales as f5/3, characterizes scalar spectra at high Reynolds numbers [22,23], and a line that follows this
scaling is shown as reference. Figure 2A shows that the spectra along the centerline exhibit only a small inertial subrange (if any), likely because the local Reynolds number at all locations is only about 2000. A large inertial-subrange is a wellknown signature of fully developed turbulence, and measurements of mixture fraction fluctuations in non-reacting flows clearly show this range diminishing with decreasing Reynolds number [23]. The farthest upstream location seems to exhibit a more extended inertial subrange, but that spectrum is not corrected for noise and that seems to contribute to the apparent power-law dependence. Figure 2A also shows that the spectra collapse relatively well in the dissipation range near f/ fB = 1. This is significant because it suggests that the estimate of the Batchelor scale is largely correct, and that the dissipation range is apparently resolved. Note, however, that no dissipation range is observed at x/d = 40 because the sampling frequency was not high enough to resolve it. This conclusion is consistent with Table 1. We also note that when scaled by the integral-scale frequency (DU/d), the spectra exhibit a roll-off at a non-dimensional frequency of about 10, which is consistent with measurements of mixture fraction fluctuations in non-reacting jets [23]. However, in agreement with non-reacting jet studies, the dissipation range could not be collapsed with this outer-scale normalization of the frequency. This latter result is different from that of [21], whose OH fluctuation spectra could be collapsed over the entire frequency range with integral-scale normalization. Figure 2B shows the power spectra at the same downstream locations but along the ray where r/ d = 0.4, which is close to the region of maximum shear. The convective Batchelor frequency used in the normalization of the frequency was scaled by the local velocity. Figure 2B shows that, in contrast to the centerline locations, the spectra seem to exhibit an extended inertial-subrange. This observation of a larger power-law region for increasing radial location is similar to what has been seen in non-reacting jet scalar dissipation measurements [24], and is likely related to the proximity to the peak shear region. This similarity with non-reacting jets is somewhat surprising at the x/d = 40 location, because the r/d = 0.4 location is near the reaction zone, and so substantial turbulence damping by the increased viscosity is expected. Figure 2C shows normalized temperature power spectra as a function of radial location at the x/d = 80 station, which is fully resolved. Note that this station is past the stoichiometric flame length and point of maximum centerline temperature. Although not entirely apparent from the figure, the spectra exhibit an increasing inertialsubrange with increasing radial location, and they collapse relatively well in the dissipation range.
G.H. Wang et al. / Proceedings of the Combustion Institute 30 (2005) 691–699
Since this location is downstream of the stoichiometric flame tip, it represents an effectively non-reacting flow, and so its characteristics should be similar to those of a low Reynolds number, non-reacting, low-density jet. Figure 3 shows PDFs of the normalized temperature gradients in the radial direction at the jet centerline. At x/d = 60, 70, and 80, the semilog plot shows that the PDFs are similar, which indicates the similarity of the gradients when normalized by the TC/kB. All the distributions appear to exhibit approximate exponential scaling of the tails (which should appear as a straight line in the semi-log plot). Exponential scaling is well documented in non-reacting turbulent flows and is a result of the intermittent nature of the scalar dissipation fluctuations [25]. Figure 3 also shows the gradient PDFs at the x/d = 40 and 50 stations. One can see that the shapes of the radial components are similar to those for x/d > 60, but the normalized values are much smaller. The dissipation scale should be resolved for x/d > 40 (and well resolved at 60), and so it seems unlikely that inadequate resolution is the cause of the difference. Instead, the difference in the distributions may result from a difference in the nature of the gradient fluctuations or possibly the estimate of the Batchelor scale is not correct for the upstream stations. PDFs of the normalized thermal dissipation rate, which were computed from measurements made along the jet centerline and at three axial stations, are shown in Fig. 4. The PDFs were computed by using the radial term only, and the radial and axial terms. PDFs of the 2-D scalar [26] and thermal [8] dissipation have been shown to be approximately log-normal, and some evidence suggests the 1-D dissipation PDF is also log-normal [9,11]. It is seen from Fig. 4A that the PDFs that used the radial term only do not exhibit a log-normal distribution, which would appear as an inverted parabola on a log–log plot. This observation is consistent for all three axial stations. The high-dissipation values are apparently
Fig. 3. Probability density functions of the normalized radial temperature gradients along the jet centerline.
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Fig. 4. Probability density functions of temperature dissipation. (A) Log–log plot showing the effect of using radial and axial components to compute dissipation, (B) variation with radial location at x/d = 80, and (C) variation with radial location at x/d = 40.
log-normally distributed but not the low-dissipation values. However, when the axial component is included, the PDFs approach the log-normal distribution. These observations are consistent with previous measurements of scalar dissipation in non-reacting jets [22] and mixture fraction dissipation in jet flames [7]. In non-reacting flows, the 1-D dissipation exhibits a slope of 1/2 in the lowdissipation portion of the PDF when plotted in log–log coordinates, which is similar to what is seen in Fig. 4A. The power-law dependence of the low-dissipation portion of the PDF is a direct result of the 1-D gradient overestimating the total gradient vector magnitude [23]. It is interesting that including the axial gradient term improves the log-normality of the PDF as well as it does, considering that the axial term is not fully resolved at the upstream stations. It is likely that the low-dissipation end of the distribution is, in fact, better resolved because the associated structures are much larger than the Batchelor scale.
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Figure 4B shows a linear-log plot of the PDFs of thermal dissipation computed from the twocomponent data at several radial locations at x/ d = 80. This figure shows that the PDFs exhibit an essentially log-normal behavior on centerline and near the outside edge of the jet, but not at intermediate locations. For example, at r/ d = 1.0, the profile is almost bimodal, suggesting that the dissipation is either high or low at that location. The PDF at r/d = 1.0 at the x/d = 40 station is similarly not log-normal, and the shape suggests a ‘‘double-hump’’ structure. The bimodal structure results from the intermittent edge of the jet, which alternatively brings turbulent fluid or co-flow air into the probe volume. The radial distribution of the mean thermal dissipation, computed from the radial temperature gradient only, is shown in Fig. 5. The dissipation rates, shown at three axial stations, have been scaled by (TC/kB)2. Figure 5 shows that at x/ d = 40, the mean dissipation exhibits a peak away from the centerline. The off-centerline maximum is similar to what was measured in [8], but the peak mean dissipation was found to be approximately five times the centerline value, as compared to a factor of 2 in the current study. The reason for this difference is probably because their measurements were made farther upstream (relative to the flame length) than in the current study. Furthermore, radial profiles similar to the current x/ d = 40 case were obtained with dual-thermocouples in the near field of a lifted propane flame [9]. They showed that the peak values were about a factor of 2–3 larger than the centerline values, but these results should be viewed with some caution because the measurements significantly under-resolved the Batchelor scale. Figure 5 shows further that at x/d = 60, the profile may exhibit a weak local maximum off-centerline, but at x/ d = 80, the mean dissipation is clearly at a maximum on centerline. Note that measurements made in the far-field of non-reacting jets seem to show somewhat contradictory trends for the radial var-
Fig. 5. Radial profiles of the normalized mean thermal dissipation (radial component only). Measurements were made at three downstream locations: x/d = 40, 60, and 80.
iation of the mean scalar dissipation. For example, models suggest that the mean scalar dissipation should peak off-centerline near the region of maximum shear, and hence maximum turbulence intensity. Indeed, the measurements of [24], taken at a single axial station, seem to validate this. However, other measurements [26–28] show the far-field mean dissipation reaching a maximum on the centerline, similar to the thermal dissipation profiles in Fig. 5. It is not entirely clear therefore whether the jet flame exhibits different mixing characteristics than a non-reacting jet, but at the very least it can be stated that the jet flame apparently does not exhibit self-similar behavior in terms of the mean dissipation. Because the off-centerline peak in the thermal dissipation is present only at the upstream location, this suggests that it is related to the presence of the reaction zone, rather than the region of maximum shear. The reaction zone is a source of temperature fluctuations, and so perhaps the local maximum is not surprising, but this issue is worth looking at in more detail. According to [8], the thermal dissipation differs from the scalar dissipation (based on mixture fraction fluctuations) only by the factor (dT/dn)2 . The relationship between T and n for this flame is known [14–17], and it is similar to a piecewise linear function that peaks near the stoichiometric mixture fraction, and is zero at n = 0 and 1. This means that on each side of the stoichiometric mixture fraction, dT/dn is approximately constant, and hence the thermal dissipation should be proportional to the mixture fraction dissipation. Near stoichiometric, dT/dn should decrease, and so the thermal dissipation should be smaller than the mixture fraction dissipation. This argument suggests that if the underlying mean scalar dissipation peaks on centerline, and decreases with increasing radius, then the thermal dissipation would not exhibit a local maximum off-centerline, but, in fact, would exhibit a local minimum at the reaction zone. Since this is not the behavior we see, it suggests that the underlying scalar dissipation indeed exhibits an off-centerline peak. Of course, this does not explain why the downstream profiles exhibit a centerline peak but it may have to do with variations in the function dT/dn near stoichiometric. As a final point regarding Fig. 5, the dissipation was computed with the radial gradient only, and laminarization by the reaction zone may render this the dominant component of the dissipation, in contrast to the centerline where the gradients are likely to be more isotropic. It is therefore possible that the total dissipation does indeed peak at the centerline, even though the radial dissipation does not. This would be true for the upstream locations where the reaction zone is well off the centerline, but not for locations past the stoichiometric flame tip, such as at x/d = 80.
G.H. Wang et al. / Proceedings of the Combustion Institute 30 (2005) 691–699
The variation of the centerline mean thermal dissipation is shown in Fig. 6. The mean thermal dissipation is seen to increase approximately linearly from x/d = 40 to 60, reach a maximum at x/ d = 60, and then decrease linearly for x/d > 60. In [29], it is argued that in non-reacting round jets the mean scalar dissipation rate should scale as Ævæ (Ænæ/kB)2, and since Ænæ x1 in round jets and kB / dRe3=4 / x (since d x and Red is cond stant), then we have Ævæ x4. Although the measurements are not all consistent, the highly resolved dissipation data of [22] seem to validate this scaling law in non-reacting jets. The results in Fig. 6 clearly do not follow this scaling law, which was also proposed to apply to jet flames [29]. This means that either the underlying scalar dissipation decay is different in jet flames or the thermal dissipation does not reflect the underlying scalar dissipation. Once again, however, since we are using only the radial component to compute the dissipation (because it is resolved over a wider range), we cannot rule out the possibility that non-isotropy of the dissipation scales could account for the trend seen in Fig. 6. This seems unlikely, however, because any laminarization that would occur near the stoichiometric flame tip should favor the axial temperature gradient, and thus reduce the dissipation based only on the radial-gradient. One thing to note is that from the jet exit to a location somewhat upstream of the stoichiometric flame length (near x/d = 60), the factor dT/dn should be approximately constant, and so the thermal dissipation should exhibit the (x/d)4 scaling if the scalar dissipation does. We can explore this further by considering the scaling of the thermal dissipation. As above, it can be argued that the thermal dissipation rate along the jet flame centerline will scale as ÆvTæ a (ÆTCæ/kB)2. Table 1 shows that the Reynolds number is approximately constant from x/d = 40 to 60, and so the Batchelor scale should scale the same as in a non-reacting jet, i.e., kB / dRe3=4 / x. If we make the approxid mation that ÆTCæ x (which is really only true for
Fig. 6. Variation of the mean thermal dissipation (radial component only) along the jet flame centerline.
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locations upstream of the point of maximum temperature), and we make further approximation that a ÆTCæ1.8 x1.8, then we find ÆvTæ x1.8. The current data suggest a somewhat weaker dependence on x (ÆvTæ x) upstream of the location of maximum temperature on centerline. One reason for the difference between the theoretical and measured scalings may be that the assumption that ÆTCæ x is not valid from x/d = 40 to 60 because the temperature peaks near x/d = 60, and so is rolling off above x/d = 40; therefore, the mean temperature will exhibit a weaker dependence on x, and this will lead to a weaker scaling of ÆvTæ. Another possibility is that the assumed state relationship, T = T (n), may not hold in all regions of the flame or may be different from that assumed. More measurements and additional analyses are clearly warranted to clarify these issues.
5. Conclusions High-repetition rate laser Rayleigh scattering was used to study the temperature fluctuations, gradients, and thermal dissipation rate characteristics in a non-premixed turbulent jet flame at a Reynolds number of 15,200. The flame studied was similar to the TNF Workshop ‘‘DLR_A’’ simple jet flame. The radial temperature gradients are measured by two-point detection, whereas the axial gradient is measured from the temperature time-series combined with Taylors hypothesis. These two-point temperature data were used to obtain temperature power spectra and detailed statistics of the thermal dissipation rate. The temperature power spectra exhibit good collapse in the dissipation range when scaled by the convective Batchelor frequency. The off-centerline spectra (near the location of maximum shear) exhibit an inertial subrange, but little or no inertial subrange is observed on centerline. The latter observation likely results from the low local Reynolds number (Red 2000) of the jet flame. Probability density functions of the thermal dissipation are shown to deviate from log-normal in the low-dissipation portion of the distribution when only one component of the gradient is used to compute the thermal dissipation. In contrast, nearly log-normal distributions are obtained along the centerline when both axial and radial components are included. Interestingly, this is true even for axial locations where the axial gradient is not resolved. It was also shown that the thermal dissipation PDFs off centerline deviate from log-normal. This is probably due to large-scale intermittency, which biases the PDFs with the low-dissipation values associated with the co-flow air. The radial profile of the mean thermal dissipation at one-half the visible flame length exhibits a peak off centerline, whereas further downstream the peak dissipation occurs on centerline. The mean thermal dissipa-
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tion on centerline is observed to scale approximately linearly with x from x/d = 40 to 60, reach a peak at x/d = 60, and then decrease linearly from x/d = 60 to 80. This scaling of the thermal dissipation is not consistent with expected scaling laws for the scalar dissipation in non-reacting jets. A scaling law is derived for the thermal dissipation rate that accounts for heat release effects and predicts ÆvTæ x1.8, which is stronger than the measured scaling law, likely because the assumed mean temperature dependence is not correct over the measurement range. Taken as a whole, these results show that the time- and space-varying temperature field of the flame exhibits strong similarities but also clear differences with the conserved scalar field in non-reacting jets. The reasons for the observed differences are not known at this time, but they do indicate that heat release has a strong influence on the nature of the turbulent fluctuations.
Acknowledgment This work was funded by the National Science Foundation under Grant CTS-9977481.
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Comments Dirk Geyer, TU Darmstadt, Germany. Your axial thermal dissipation depends on the local mean velocity. In turbulent flames the instantaneous axial velocity will vary over a range. How critical is that for your data?
Reference [1] Mi, Antonia, Phys. Fluids 6 (1994) 1548–1552. d
Reply. We use the conventional form of Taylors hypothesis, which is based on the mean velocity. Studies in non-reacting jets have shown that this form of Taylors hypothesis will give an error in the scalar dissipation rate of less than 10% on the jet centerline but somewhat larger errors off centerline [1]. Whether the same magnitude errors occur in flames is not known, but it is likely that they are similar because the turbulence intensities ([17] in paper) are similar to those in non-reacting jets.
Stephen Pope, Cornell University, USA. In regions of the flame containing both rich and lean mixtures, the mixture fraction is a multi-valued function of temperature and hence it is not possible to infer directly the scalar dissipation. However, given a modeled form of the joint PDF of mixture fraction and scalar dissipation, it is possible to obtain the joint PDF of temperature and thermal dissipation. Have you considered this approach for (indirectly) estimat-
G.H. Wang et al. / Proceedings of the Combustion Institute 30 (2005) 691–699 ing the joint PDF of mixture fraction and scalar dissipation? Reply. This question is in response to a point that was emphasized more in the presentation than in the paper. We commented that since scalar (mixture fraction) dissipation measurements are notoriously difficult to make in flames, an alternative strategy may be to use high quality measurements of the thermal dissipation to infer certain characteristics of the underlying scalar dis-
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sipation field. For example, along the jet flame centerline, well away from the stoichiometric flame tip, the mapping from thermal to scalar dissipation is relatively straightforward because the state relationship between temperature and mixture fraction can be unambiguously determined. However, Professor Pope makes an excellent suggestion that could make the thermal dissipation measurements more generally useful for combustion model validation. We thank him for his suggestion.
Combustion Institute
Proceedings of the Combustion Institute 30 (2005) 691–699
www.elsevier.com/locate/proci
High-repetition rate measurements of temperature and thermal dissipation in a non-premixed turbulent jet flame G.H. Wang, N.T. Clemens*, P.L. Varghese Department of Aerospace Engineering and Engineering Mechanics, Center for Aeromechanics Research, The University of Texas at Austin, Austin, TX 78712-1085, USA
Abstract High-repetition rate laser Rayleigh scattering is used to study the temperature fluctuations, power spectra, gradients, and thermal dissipation rate characteristics of a non-premixed turbulent jet flame at a Reynolds number of 15,200. The radial temperature gradient is measured by a two-point technique, whereas the axial gradient is measured from the temperature time-series combined with Taylors hypothesis. The temperature power spectra along the jet centerline exhibit only a small inertial subrange, probably because of the low local Reynolds number (Red 2000), although a larger inertial subrange is present in the spectra at off-centerline locations. Scaling the frequency by the estimated Batchelor frequency improves the collapse of the dissipation region of the spectra, but this collapse is not as good as is obtained in non-reacting jets. Probability density functions of the thermal dissipation are shown to deviate from lognormal in the low-dissipation portion of the distribution when only one component of the gradient is used. In contrast, nearly log-normal distributions are obtained along the centerline when both axial and radial components are included, even for locations where the axial gradient is not resolved. The thermal dissipation PDFs measured off the centerline deviate from log-normal owing to large-scale intermittency. At one-half the visible flame length, the radial profile of the mean thermal dissipation exhibits a peak off the centerline, whereas farther downstream the peak dissipation occurs on the centerline. The mean thermal dissipation on centerline is observed to increase linearly with downstream distance, reach a peak at the location of maximum mean centerline temperature, and then decrease for farther downstream locations. Many of these observed trends are not consistent with equivalent non-reacting turbulent jet measurements, and thus indicate the importance of understanding how heat release modifies the turbulence structure of jet flames. 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Turbulent flame; Scalar dissipation; Rayleigh scattering
1. Introduction Detailed measurements of mean and fluctuating scalars, such as species mass fractions and *
Corresponding author. Fax: +1 512 471 3788. E-mail address: [email protected] (N.T. Clemens).
temperature, have been critical to developing an improved understanding of the physics of turbulent non-premixed flames [1–4]. Of particular importance are the mixture fraction n and its gradients, because in the flamelet theory the flame structure is fundamentally related to the value of n at stoichiometric conditions and the rate of scalar dissipation, v ” 2D (,n Æ ,n), where D is the
1540-7489/$ - see front matter 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2004.08.269
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diffusivity. The scalar dissipation rate, which is a measure of the mixing rate, limits the reaction rate under mixing limited conditions, and affects the degree of non-equilibrium under finite chemistry conditions. Because of its importance to combustion, measurements of v in turbulent non-premixed flames have received a lot of attention in recent years [5–7], but detailed statistical measurements of the type that exist for non-reacting flows are still relatively sparse. It can be argued that temperature does not play as fundamental a role as mixture fraction in determining the flame characteristics, but its fluctuations and gradients do provide important information about the underlying mixture fraction structure. This can be seen by considering the thermal dissipation rate vT = 2a (,T Æ ,T), where T is the temperature and a is the thermal diffusivity, which is related to the rate of thermal mixing, or alternatively to the rate at which thermal inhomogeneities are removed by diffusion. Importantly, under the assumption of the state relationship T = T (n) and unity Lewis number, the scalar and thermal dissipation rates are related as vT = v (dT/dn)2 [8]. As will be discussed in Section 4, in some regions of the flame, dT/dn is approximately constant, and so the thermal dissipation rate is proportional to the scalar dissipation rate. Thermal mixing is also important because it affects high-temperature chemical reaction processes and can be important in the development and validation of turbulent flame models [8]. For these reasons, fluctuating temperature and thermal dissipation rates have been measured in a number of studies by using, e.g., dual-thermocouple measurements [9,10], two-point laser Rayleigh measurements [11,12], and planar laser Rayleigh imaging [8]. An important issue with such measurements is that the requirement to obtain fully spatially and temporally resolved measurements of the finest scales of turbulence is very stringent and this makes dissipation measurements, in particular, challenging [13]. It is clear from a careful study of the literature that the Batchelor scale is rarely resolved, even in turbulent flame studies that explicitly seek to measure the dissipation rate. The objective in this study was to make highquality, high-repetition rate (10 kHz), two-point laser Rayleigh temperature measurements in a weakly co-flowing turbulent non-premixed jet flame at a Reynolds number of 15,200, with high signal-to-noise ratio (50 in room air) and where the finest scales of turbulence are spatially and temporally resolved. These two-point temperature data were used to obtain temperature power spectra and detailed statistics of the thermal dissipation rate. The flame studied here is similar to the TNF simple jet flame (DLR_A), which is used as a benchmark flame for the TNF Workshop [14–17].
2. Experimental setup The flow studied was a weakly co-flowing jet flame. The co-flow air was filtered to remove particles larger than 0.2 lm and then passed through a flow conditioning section. The co-flow velocity was 0.45 m/s. The fuel issued from a long tube with inside diameter d = 7.75 mm. The test section had a 0.75 m · 0.75 m cross-section. The whole jet flow facility was mounted on a traverse that was driven by stepper motors to provide positioning in the radial and axial directions. The fuel composition used in this study was 22.1% CH4, 33.2% H2, and 44.7% N2 (by volume), which gives a stoichiometric mixture fraction of 0.167. The fuel gases were metered by pressure regulators and monitored by mass flowmeters to an accuracy of ±1.0 SLPM for N2 and ±0.5 SLPM for CH4 and H2. The Rayleigh cross-section of this fuel has been shown to vary by ±3% across the whole flame [15]. The source Reynolds number was Red = U0d/m0 = 15,200 (where m0 is the kinematic viscosity of the fuel and U0 is the jet exit bulk velocity), and the measurements were taken at downstream locations from x/d = 40 to 80. Here, x and r are the axial and radial coordinates, respectively. The visible flame length was at about x/d = 84, and the stoichiometric flame length, estimated based on data in the TNF database, was at about x/d = 60. The laser Rayleigh system is based on a diodepumped Nd:YAG laser operated at 71 W average power at 532 nm and with a 10 kHz repetition rate. The laser beam was focused into the test section by using a 300 mm focal length lens. The beam diameter was measured to be about 0.3 mm. An external photodiode was used to correct for variations in the laser pulse energy on a shot-by-shot basis. Rayleigh scattered light was collected using custom-designed optics that consisted of a pair of 150 mm diameter plano-convex lenses, one 50.8 mm diameter meniscus lens and one 50.8 mm diameter double-convex lens. The lens system was designed with ZEMAX and produced an aberration-limited blur-spot of less than 34 lm. The working f# was 2.4, and the magnification was 0.685. Two-point measurements were made by imaging the scattered light onto a broadband hybrid cube beam splitter, which reflected and transmitted the split signal onto two different PMTs. Two 200 lm slits were placed in front of the PMTs to define the spatial resolution (i.e., length of the beam imaged). The slit width in the image plane corresponded to 300 lm in the object plane. These two slits were arranged such that the separation of probe volumes in the flow was 300 lm. The PMT and photodiode outputs were read by gated integrators operated with gate widths of 300 ns. The integrated signals were synchronously sampled by a 12-bit A/D converter at 10 kHz.
G.H. Wang et al. / Proceedings of the Combustion Institute 30 (2005) 691–699
For constant Rayleigh cross-section, the temperature is derived from the formula: T ¼ I R;ref T ref =I R , where I R;ref is the reference Rayleigh scattering signal from air at room temperature (Tref). The two-point instantaneous temperature signals were used to determine the instantaneous radial temperature gradient by using the approximation oT/or = DT/Dr, where DT is the temperature difference of the two measurements and Dr = 300 lm is the probe separation distance. The axial gradient is estimated by the time-series and Taylors hypothesis, oT/ ox = (1/U) DT/Dt, where U is the local velocity obtained from [17]. The error in using Taylors hypothesis is estimated to be about 10% at the jet centerline [11,18]. For convenience, we define the thermal dissipation based on single components of the gradient vector: vT ;r ¼ 2aðoT =orÞ2 , and vT ;2D ¼ vT ;r þ vT ;x , vT ;x ¼ 2aU 2 ðoT =otÞ2 , where the thermal diffusivity is computed from the formula a = 2 · 105 (T/300)1.8 (m2/s) [11]. 3. Resolution estimation The finest spatial structures in the scalar field are of the order of the Batchelor scale, which is defined as kB = g Sc1/2, where g = (m3/Æeæ)1/4 is the Kolmogorov scale, m is the kinematic viscosity, Æeæ is the mean rate of kinetic energy dissipation, and Sc = m/D is the Schmidt number. Using measurements of Æeæ in non-reacting round jets [19], the Batchelor scale can be shown to be equal to kB = 2.3 d Red3/4 Sc1/2, where Red is the local Reynolds number and is defined as UCd/m, with UC the jet centerline velocity and d the full width at half maximum of the velocity profile. In non-reacting jets, Red is approximately equal to Red but this is not true in jet flames [20]. For example, the local Reynolds numbers at the downstream stations studied are shown in Table 1. The kinematic viscosity was estimated to be that of air at the measured centerline temperature according to the formula m = m0 (TC/T0)1.7. For time-series measurements, the highest frequency present in the flow is expected to be the ‘‘convective’’ Batchelor frequency fB = U/(2pkB), where U is the local mean velocity. Note that the Batchelor frequency can be quite large in jet
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flames and is not always resolved because signalto-noise ratio considerations usually necessitate relatively low bandwidths. To estimate fB in this study, the velocities used were taken from [17]. The velocities and the estimated Batchelor scales and Batchelor frequencies are shown in Table 1 for various locations along the jet centerline. Off-centerline mean velocities were also taken from [17] and used for scaling the data where appropriate, but these values are not shown in the table. Table 1 shows that the current twopoint spatial separation of 300 lm is equal to kB at x/d = 40 and is about a factor of 2 smaller than kB farther downstream. This suggests that the radial gradient measurement is fully spatially resolved. The convective Batchelor frequency fB is about 2 times the frequency resolution at x/ d = 40, and the errors in the axial dissipation are expected to be about 10%. The sampling frequency of this study is sufficient to resolve the convective Batchelor frequency (and hence the Batchelor scale) along the centerline for x/ d P 60. The measurements were limited to stations of x/d = 40 and larger so that the resolution of the dissipation scales could be maintained, at least for the radial gradient. 4. Results and discussion Figure 1 shows the measured mean and rms radial temperature profiles at three downstream stations (x/d = 40, 60, and 80). For comparison, the temperature profiles from the TNF database [14] are also shown, and it is seen that the current measurements agree very well with the database. Temperature power spectra are shown in Fig. 2. In Fig. 2A, spectra are shown along the centerline of the jet flame at the five axial stations in the range x/d = 40–80. The frequency is normalized by the convective Batchelor frequency, and the power spectral density is normalized by T 2rms =fB . The spectra have also been corrected to remove the contribution from shot-noise by using the procedure developed in [21]. This procedure requires that the measurement cut-off frequency be higher than the Batchelor frequency so that the power of the noise fluctuations can be determined. Since this was not the case for all measure-
Table 1 Jet flame conditions and estimates of resolution requirements x/d
TC (K)
U C =U C0
d/d
Red
kB (mm)
fB (kHz)
40 50 60 70 80
1580 1745 1840 1680 1470
0.36 0.29 0.24 0.21 0.18
5.3 6.7 8.0 9.3 10.6
2090 1760 1610 1870 2350
0.32 0.45 0.58 0.60 0.58
9.3 5.3 3.4 2.8 2.5
Velocity data from [17], U C0 ¼ 51:6 m=s is the mean jet exit centerline velocity, which is different from the jet exit bulk velocity U0; temperature data from TNF database [14].
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Fig. 1. Radial profiles of (A) mean and (B) rms temperature at three different axial stations (x/d = 40, 60, and 80). Data from TNF workshop database [14] are shown for comparison.
Fig. 2. Fluctuating temperature power spectra. (A) Spectra at the jet flame centerline (corrected except at x/d = 40 and 50); (B) along ray r/d = 0.4 (corrected except at x/d = 40); and (C) at x/d = 80 and different radial locations, all corrected.
ment locations, the correction was not applied in all cases (see figure caption). All of the spectra exhibit a similar appearance in that they are relatively flat at low frequency, begin rolling off for f/fB > 102, and then roll-off more rapidly in the dissipation range (f/fB > 0.4). An inertial subrange, where the power scales as f5/3, characterizes scalar spectra at high Reynolds numbers [22,23], and a line that follows this
scaling is shown as reference. Figure 2A shows that the spectra along the centerline exhibit only a small inertial subrange (if any), likely because the local Reynolds number at all locations is only about 2000. A large inertial-subrange is a wellknown signature of fully developed turbulence, and measurements of mixture fraction fluctuations in non-reacting flows clearly show this range diminishing with decreasing Reynolds number [23]. The farthest upstream location seems to exhibit a more extended inertial subrange, but that spectrum is not corrected for noise and that seems to contribute to the apparent power-law dependence. Figure 2A also shows that the spectra collapse relatively well in the dissipation range near f/ fB = 1. This is significant because it suggests that the estimate of the Batchelor scale is largely correct, and that the dissipation range is apparently resolved. Note, however, that no dissipation range is observed at x/d = 40 because the sampling frequency was not high enough to resolve it. This conclusion is consistent with Table 1. We also note that when scaled by the integral-scale frequency (DU/d), the spectra exhibit a roll-off at a non-dimensional frequency of about 10, which is consistent with measurements of mixture fraction fluctuations in non-reacting jets [23]. However, in agreement with non-reacting jet studies, the dissipation range could not be collapsed with this outer-scale normalization of the frequency. This latter result is different from that of [21], whose OH fluctuation spectra could be collapsed over the entire frequency range with integral-scale normalization. Figure 2B shows the power spectra at the same downstream locations but along the ray where r/ d = 0.4, which is close to the region of maximum shear. The convective Batchelor frequency used in the normalization of the frequency was scaled by the local velocity. Figure 2B shows that, in contrast to the centerline locations, the spectra seem to exhibit an extended inertial-subrange. This observation of a larger power-law region for increasing radial location is similar to what has been seen in non-reacting jet scalar dissipation measurements [24], and is likely related to the proximity to the peak shear region. This similarity with non-reacting jets is somewhat surprising at the x/d = 40 location, because the r/d = 0.4 location is near the reaction zone, and so substantial turbulence damping by the increased viscosity is expected. Figure 2C shows normalized temperature power spectra as a function of radial location at the x/d = 80 station, which is fully resolved. Note that this station is past the stoichiometric flame length and point of maximum centerline temperature. Although not entirely apparent from the figure, the spectra exhibit an increasing inertialsubrange with increasing radial location, and they collapse relatively well in the dissipation range.
G.H. Wang et al. / Proceedings of the Combustion Institute 30 (2005) 691–699
Since this location is downstream of the stoichiometric flame tip, it represents an effectively non-reacting flow, and so its characteristics should be similar to those of a low Reynolds number, non-reacting, low-density jet. Figure 3 shows PDFs of the normalized temperature gradients in the radial direction at the jet centerline. At x/d = 60, 70, and 80, the semilog plot shows that the PDFs are similar, which indicates the similarity of the gradients when normalized by the TC/kB. All the distributions appear to exhibit approximate exponential scaling of the tails (which should appear as a straight line in the semi-log plot). Exponential scaling is well documented in non-reacting turbulent flows and is a result of the intermittent nature of the scalar dissipation fluctuations [25]. Figure 3 also shows the gradient PDFs at the x/d = 40 and 50 stations. One can see that the shapes of the radial components are similar to those for x/d > 60, but the normalized values are much smaller. The dissipation scale should be resolved for x/d > 40 (and well resolved at 60), and so it seems unlikely that inadequate resolution is the cause of the difference. Instead, the difference in the distributions may result from a difference in the nature of the gradient fluctuations or possibly the estimate of the Batchelor scale is not correct for the upstream stations. PDFs of the normalized thermal dissipation rate, which were computed from measurements made along the jet centerline and at three axial stations, are shown in Fig. 4. The PDFs were computed by using the radial term only, and the radial and axial terms. PDFs of the 2-D scalar [26] and thermal [8] dissipation have been shown to be approximately log-normal, and some evidence suggests the 1-D dissipation PDF is also log-normal [9,11]. It is seen from Fig. 4A that the PDFs that used the radial term only do not exhibit a log-normal distribution, which would appear as an inverted parabola on a log–log plot. This observation is consistent for all three axial stations. The high-dissipation values are apparently
Fig. 3. Probability density functions of the normalized radial temperature gradients along the jet centerline.
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Fig. 4. Probability density functions of temperature dissipation. (A) Log–log plot showing the effect of using radial and axial components to compute dissipation, (B) variation with radial location at x/d = 80, and (C) variation with radial location at x/d = 40.
log-normally distributed but not the low-dissipation values. However, when the axial component is included, the PDFs approach the log-normal distribution. These observations are consistent with previous measurements of scalar dissipation in non-reacting jets [22] and mixture fraction dissipation in jet flames [7]. In non-reacting flows, the 1-D dissipation exhibits a slope of 1/2 in the lowdissipation portion of the PDF when plotted in log–log coordinates, which is similar to what is seen in Fig. 4A. The power-law dependence of the low-dissipation portion of the PDF is a direct result of the 1-D gradient overestimating the total gradient vector magnitude [23]. It is interesting that including the axial gradient term improves the log-normality of the PDF as well as it does, considering that the axial term is not fully resolved at the upstream stations. It is likely that the low-dissipation end of the distribution is, in fact, better resolved because the associated structures are much larger than the Batchelor scale.
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Figure 4B shows a linear-log plot of the PDFs of thermal dissipation computed from the twocomponent data at several radial locations at x/ d = 80. This figure shows that the PDFs exhibit an essentially log-normal behavior on centerline and near the outside edge of the jet, but not at intermediate locations. For example, at r/ d = 1.0, the profile is almost bimodal, suggesting that the dissipation is either high or low at that location. The PDF at r/d = 1.0 at the x/d = 40 station is similarly not log-normal, and the shape suggests a ‘‘double-hump’’ structure. The bimodal structure results from the intermittent edge of the jet, which alternatively brings turbulent fluid or co-flow air into the probe volume. The radial distribution of the mean thermal dissipation, computed from the radial temperature gradient only, is shown in Fig. 5. The dissipation rates, shown at three axial stations, have been scaled by (TC/kB)2. Figure 5 shows that at x/ d = 40, the mean dissipation exhibits a peak away from the centerline. The off-centerline maximum is similar to what was measured in [8], but the peak mean dissipation was found to be approximately five times the centerline value, as compared to a factor of 2 in the current study. The reason for this difference is probably because their measurements were made farther upstream (relative to the flame length) than in the current study. Furthermore, radial profiles similar to the current x/ d = 40 case were obtained with dual-thermocouples in the near field of a lifted propane flame [9]. They showed that the peak values were about a factor of 2–3 larger than the centerline values, but these results should be viewed with some caution because the measurements significantly under-resolved the Batchelor scale. Figure 5 shows further that at x/d = 60, the profile may exhibit a weak local maximum off-centerline, but at x/ d = 80, the mean dissipation is clearly at a maximum on centerline. Note that measurements made in the far-field of non-reacting jets seem to show somewhat contradictory trends for the radial var-
Fig. 5. Radial profiles of the normalized mean thermal dissipation (radial component only). Measurements were made at three downstream locations: x/d = 40, 60, and 80.
iation of the mean scalar dissipation. For example, models suggest that the mean scalar dissipation should peak off-centerline near the region of maximum shear, and hence maximum turbulence intensity. Indeed, the measurements of [24], taken at a single axial station, seem to validate this. However, other measurements [26–28] show the far-field mean dissipation reaching a maximum on the centerline, similar to the thermal dissipation profiles in Fig. 5. It is not entirely clear therefore whether the jet flame exhibits different mixing characteristics than a non-reacting jet, but at the very least it can be stated that the jet flame apparently does not exhibit self-similar behavior in terms of the mean dissipation. Because the off-centerline peak in the thermal dissipation is present only at the upstream location, this suggests that it is related to the presence of the reaction zone, rather than the region of maximum shear. The reaction zone is a source of temperature fluctuations, and so perhaps the local maximum is not surprising, but this issue is worth looking at in more detail. According to [8], the thermal dissipation differs from the scalar dissipation (based on mixture fraction fluctuations) only by the factor (dT/dn)2 . The relationship between T and n for this flame is known [14–17], and it is similar to a piecewise linear function that peaks near the stoichiometric mixture fraction, and is zero at n = 0 and 1. This means that on each side of the stoichiometric mixture fraction, dT/dn is approximately constant, and hence the thermal dissipation should be proportional to the mixture fraction dissipation. Near stoichiometric, dT/dn should decrease, and so the thermal dissipation should be smaller than the mixture fraction dissipation. This argument suggests that if the underlying mean scalar dissipation peaks on centerline, and decreases with increasing radius, then the thermal dissipation would not exhibit a local maximum off-centerline, but, in fact, would exhibit a local minimum at the reaction zone. Since this is not the behavior we see, it suggests that the underlying scalar dissipation indeed exhibits an off-centerline peak. Of course, this does not explain why the downstream profiles exhibit a centerline peak but it may have to do with variations in the function dT/dn near stoichiometric. As a final point regarding Fig. 5, the dissipation was computed with the radial gradient only, and laminarization by the reaction zone may render this the dominant component of the dissipation, in contrast to the centerline where the gradients are likely to be more isotropic. It is therefore possible that the total dissipation does indeed peak at the centerline, even though the radial dissipation does not. This would be true for the upstream locations where the reaction zone is well off the centerline, but not for locations past the stoichiometric flame tip, such as at x/d = 80.
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The variation of the centerline mean thermal dissipation is shown in Fig. 6. The mean thermal dissipation is seen to increase approximately linearly from x/d = 40 to 60, reach a maximum at x/ d = 60, and then decrease linearly for x/d > 60. In [29], it is argued that in non-reacting round jets the mean scalar dissipation rate should scale as Ævæ (Ænæ/kB)2, and since Ænæ x1 in round jets and kB / dRe3=4 / x (since d x and Red is cond stant), then we have Ævæ x4. Although the measurements are not all consistent, the highly resolved dissipation data of [22] seem to validate this scaling law in non-reacting jets. The results in Fig. 6 clearly do not follow this scaling law, which was also proposed to apply to jet flames [29]. This means that either the underlying scalar dissipation decay is different in jet flames or the thermal dissipation does not reflect the underlying scalar dissipation. Once again, however, since we are using only the radial component to compute the dissipation (because it is resolved over a wider range), we cannot rule out the possibility that non-isotropy of the dissipation scales could account for the trend seen in Fig. 6. This seems unlikely, however, because any laminarization that would occur near the stoichiometric flame tip should favor the axial temperature gradient, and thus reduce the dissipation based only on the radial-gradient. One thing to note is that from the jet exit to a location somewhat upstream of the stoichiometric flame length (near x/d = 60), the factor dT/dn should be approximately constant, and so the thermal dissipation should exhibit the (x/d)4 scaling if the scalar dissipation does. We can explore this further by considering the scaling of the thermal dissipation. As above, it can be argued that the thermal dissipation rate along the jet flame centerline will scale as ÆvTæ a (ÆTCæ/kB)2. Table 1 shows that the Reynolds number is approximately constant from x/d = 40 to 60, and so the Batchelor scale should scale the same as in a non-reacting jet, i.e., kB / dRe3=4 / x. If we make the approxid mation that ÆTCæ x (which is really only true for
Fig. 6. Variation of the mean thermal dissipation (radial component only) along the jet flame centerline.
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locations upstream of the point of maximum temperature), and we make further approximation that a ÆTCæ1.8 x1.8, then we find ÆvTæ x1.8. The current data suggest a somewhat weaker dependence on x (ÆvTæ x) upstream of the location of maximum temperature on centerline. One reason for the difference between the theoretical and measured scalings may be that the assumption that ÆTCæ x is not valid from x/d = 40 to 60 because the temperature peaks near x/d = 60, and so is rolling off above x/d = 40; therefore, the mean temperature will exhibit a weaker dependence on x, and this will lead to a weaker scaling of ÆvTæ. Another possibility is that the assumed state relationship, T = T (n), may not hold in all regions of the flame or may be different from that assumed. More measurements and additional analyses are clearly warranted to clarify these issues.
5. Conclusions High-repetition rate laser Rayleigh scattering was used to study the temperature fluctuations, gradients, and thermal dissipation rate characteristics in a non-premixed turbulent jet flame at a Reynolds number of 15,200. The flame studied was similar to the TNF Workshop ‘‘DLR_A’’ simple jet flame. The radial temperature gradients are measured by two-point detection, whereas the axial gradient is measured from the temperature time-series combined with Taylors hypothesis. These two-point temperature data were used to obtain temperature power spectra and detailed statistics of the thermal dissipation rate. The temperature power spectra exhibit good collapse in the dissipation range when scaled by the convective Batchelor frequency. The off-centerline spectra (near the location of maximum shear) exhibit an inertial subrange, but little or no inertial subrange is observed on centerline. The latter observation likely results from the low local Reynolds number (Red 2000) of the jet flame. Probability density functions of the thermal dissipation are shown to deviate from log-normal in the low-dissipation portion of the distribution when only one component of the gradient is used to compute the thermal dissipation. In contrast, nearly log-normal distributions are obtained along the centerline when both axial and radial components are included. Interestingly, this is true even for axial locations where the axial gradient is not resolved. It was also shown that the thermal dissipation PDFs off centerline deviate from log-normal. This is probably due to large-scale intermittency, which biases the PDFs with the low-dissipation values associated with the co-flow air. The radial profile of the mean thermal dissipation at one-half the visible flame length exhibits a peak off centerline, whereas further downstream the peak dissipation occurs on centerline. The mean thermal dissipa-
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tion on centerline is observed to scale approximately linearly with x from x/d = 40 to 60, reach a peak at x/d = 60, and then decrease linearly from x/d = 60 to 80. This scaling of the thermal dissipation is not consistent with expected scaling laws for the scalar dissipation in non-reacting jets. A scaling law is derived for the thermal dissipation rate that accounts for heat release effects and predicts ÆvTæ x1.8, which is stronger than the measured scaling law, likely because the assumed mean temperature dependence is not correct over the measurement range. Taken as a whole, these results show that the time- and space-varying temperature field of the flame exhibits strong similarities but also clear differences with the conserved scalar field in non-reacting jets. The reasons for the observed differences are not known at this time, but they do indicate that heat release has a strong influence on the nature of the turbulent fluctuations.
Acknowledgment This work was funded by the National Science Foundation under Grant CTS-9977481.
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Comments Dirk Geyer, TU Darmstadt, Germany. Your axial thermal dissipation depends on the local mean velocity. In turbulent flames the instantaneous axial velocity will vary over a range. How critical is that for your data?
Reference [1] Mi, Antonia, Phys. Fluids 6 (1994) 1548–1552. d
Reply. We use the conventional form of Taylors hypothesis, which is based on the mean velocity. Studies in non-reacting jets have shown that this form of Taylors hypothesis will give an error in the scalar dissipation rate of less than 10% on the jet centerline but somewhat larger errors off centerline [1]. Whether the same magnitude errors occur in flames is not known, but it is likely that they are similar because the turbulence intensities ([17] in paper) are similar to those in non-reacting jets.
Stephen Pope, Cornell University, USA. In regions of the flame containing both rich and lean mixtures, the mixture fraction is a multi-valued function of temperature and hence it is not possible to infer directly the scalar dissipation. However, given a modeled form of the joint PDF of mixture fraction and scalar dissipation, it is possible to obtain the joint PDF of temperature and thermal dissipation. Have you considered this approach for (indirectly) estimat-
G.H. Wang et al. / Proceedings of the Combustion Institute 30 (2005) 691–699 ing the joint PDF of mixture fraction and scalar dissipation? Reply. This question is in response to a point that was emphasized more in the presentation than in the paper. We commented that since scalar (mixture fraction) dissipation measurements are notoriously difficult to make in flames, an alternative strategy may be to use high quality measurements of the thermal dissipation to infer certain characteristics of the underlying scalar dis-
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sipation field. For example, along the jet flame centerline, well away from the stoichiometric flame tip, the mapping from thermal to scalar dissipation is relatively straightforward because the state relationship between temperature and mixture fraction can be unambiguously determined. However, Professor Pope makes an excellent suggestion that could make the thermal dissipation measurements more generally useful for combustion model validation. We thank him for his suggestion.