(To be Published in the Proceedings of the Combustion Institute, Vol. 30, 2004)
High-Repetition Rate Measurements of Temperature
and Thermal Dissipation in a Nonpremixed Turbulent
Jet Flame
G.H. Wang, N.T. Clemens* and P.L. Varghese
Center for Aeromechanics Research
Department of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
Austin, Texas 78712-1085
*Corresponding author:
Center for Aeromechanics Research
Department of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
Austin TX 78705-1085, USA
Fax:
1-512-471-3788
Email: [email protected].
PID number:
PID29879
Running Title:
Time-Series Thermal Dissipation Measurement
1
ABSTRACT
High-repetition rate laser Rayleigh scattering is used to study the
temperature fluctuations, power spectra, gradients and thermal dissipation rate
characteristics of a nonpremixed 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 Taylor’s hypothesis. The temperature power spectra along the jet centerline
exhibit only a small inertial subrange, probably because of the low local Reynolds
number (Reδ ≈ 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 nonreacting 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 lognormal 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 lognormal 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 nonreacting turbulent jet
2
measurements, and thus indicate the importance of understanding how heat release
modifies the turbulence structure of jet flames.
KEYWORDS:
TURBULENT FLAME, SCALAR DISSIPATION, RAYLEIGH SCATTERING
3
1. INTRODUCTION
Detailed measurements of mean and fluctuating scalars, such as species
mass fractions and temperature have been critical to developing an improved
understanding of the physics of turbulent nonpremixed flames [1-4]. Of particular
importance are the mixture fraction ξ and its gradients, because in flamelet theory
the flame structure is fundamentally related to the value of ξ at stoichiometric
conditions and the rate of scalar dissipation, χ ≡ 2D (∇ξ⋅∇ξ), where D is the
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
nonequilibrium under finite chemistry conditions. Because of its importance to
combustion, measurements of χ in turbulent nonpremixed flames have received a lot
of attention in recent years [5-7], but detailed statistical measurements of the type
that exist for nonreacting 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
χΤ = 2α (∇T·∇T), where T is temperature and α 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(ξ) and unity Lewis number, the scalar and thermal
dissipation rates are related as χΤ = χ (dΤ/dξ)2 [8]. As will be discussed in the
results section, in some regions of the flame, dT/dξ is approximately constant, and
so the thermal dissipation rate is proportional to the scalar dissipation rate.
4
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., dualthermocouple 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 temporallyresolved measurements of the finest scales of turbulence is very stringent and this
makes dissipation measurements particularly challenging [13]. It is clear from
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 is to make high-quality, high-repetition rate (10
kHz), two-point laser Rayleigh temperature measurements in a weakly co-flowing
turbulent nonpremixed jet flame at a Reynolds number of 15,200, with high signalto-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 coflow air was
filtered to remove particles larger than 0.2 µm and then passed through a flow
conditioning section. The coflow 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
5
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, 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 flow meters 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/ν0 = 15,200 (where ν0 is the kinematic viscosity of
the fuel and U0 is the mean jet exit 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 diode-pumped Nd:YAG laser
operated at 71W 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 µm. 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
µm slits were placed in front of the PMTs to define the spatial resolution (i.e.,
6
length of the beam imaged). The slit width in the image plane corresponded to 300
µm in the object plane. These two slits were arranged such that the separation of
probe volumes in the flow was 300 µm. 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.
For constant Rayleigh cross section, the temperature is derived from the
formula: T = IR,ref Tref / IR, where IR,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 ∂T/∂r = ∆T/∆r, where ∆T is the temperature difference of the two
measurements and ∆r = 300 µm is the probe separation distance. The axial gradient
is estimated by the time-series and Taylor’s hypothesis, ∂T/∂x = - (1/U) ∆T/∆t,
where U is the local velocity obtained from [17]. The error in using Taylor’s
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: χΤ ,r = 2α (∂Τ/∂r)2, χΤ ,x = 2α U -2 (∂Τ/∂t)2 and χΤ ,2D = χΤ ,r+χΤ ,x,
where the thermal diffusivity is computed from the formula α = 2 × 10 -5 (T/300)1.8
(m2/s) [11].
3. RESOLUTION ESTIMATION
The finest spatial structures in the scalar field are of order the Batchelor
scale, which is defined as λB = η Sc -1/2, where η = (ν 3/ <ε>) 1/4 is the Kolmogorov
scale, ν is the kinematic viscosity, <ε> is the mean rate of kinetic energy dissipation
and Sc = ν / D is the Schmidt number. Using measurements of <ε> in nonreacting
round jets [19], the Batchelor scale can be shown to be equal to
λB = 2.3 δ Reδ -3/4 Sc -1/2, where Reδ is the local Reynolds number and is defined
7
as UC δ /ν, with UC the jet centerline velocity and δ the full width at half maximum
of the velocity profile. In nonreacting jets Red is approximately equal to Reδ 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 the local fluid mixture at the measured centerline temperature
per the formula ν = ν0 (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/(2πλB), where U is the
local mean velocity. Note that the Batchelor frequency can be quite large in jet
flames and is not always resolved because signal-to-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 two-point spatial separation of 300 µm is equal to λB
at x/d = 40 and is about a factor of two smaller than λB 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 convective Batchelor
frequency (and hence the Batchelor scale) along the centerline for x/d ≥ 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.
8
4. RESULTS AND DISCUSSION
Figure 1 shows measured mean and rms radial temperature profiles at three
downstream stations (x/d = 40, 60, 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 with the range
x/d=40-80. The frequency is normalized by the convective Batchelor frequency, and
the power spectral density is normalized by Trms2/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
is higher than the Batchelor frequency so that the power of the noise fluctuations
can be determined. Since this was not the case for all of the measurement 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 > 10-2, and then roll-off more rapidly in
the dissipation range (f/fB > 0.4). An inertial subrange, where the power scales as
f -5/3, characterizes scalar spectra at high Reynolds numbers [22,23], and a line that
follows this scaling is shown as reference. Figure 2a shows 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
well known signature of fully developed turbulence and measurements of mixture
fraction fluctuations 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
9
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 (∆U/δ), the spectra exhibit a rolloff 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 nonreacting 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/δ = 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 nonreacting jets is somewhat surprising at the x/d = 40 location, because the r/δ = 0.4
location is near the reaction zone and so substantial turbulence damping by the
increased viscosity is expected.
10
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 inertial-subrange with increasing radial location, and they collapse
relatively well in the dissipation range. Since this location is downstream of the
stoichiometric flame tip it represents an effectively nonreacting flow, and so its
characteristics should be similar to those of a low Reynolds number, nonreacting,
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 semi-log plot shows that
the PDFs are similar, which indicates the similarity of the gradients when
normalized by the TC /λB. All the distributions appear to exhibit approximately
exponential scaling of the tails (which should appear as a straight line in the semilog plot). Exponential scaling is well documented in nonreacting 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
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Figure 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 log-normally
distributed but not the low dissipation values. However, when the axial component
is included, the PDFs approach the lognormal distribution. These observations are
consistent with previous measurements of scalar dissipation in nonreacting jets [22]
and mixture fraction dissipation in jet flames [7]. In nonreacting flows the 1-D
dissipation exhibits a slope of ½ in the low-dissipation 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.
Figure 4b shows a linear-log plot of the PDFs of thermal dissipation
computed from the two-component data at several radial locations at x/d = 80. This
figure shows that the PDFs exhibit essentially log-normal behavior on centerline
and near the outside edge of the jet, but not at intermediate locations. For example,
at r/δ = 1.0, the profile is almost bimodal, suggesting that the dissipation is either
12
high or low at that location. The PDF at r/δ = 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/λB)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 they found the peak mean
dissipation to be approximately five times the centerline value, as compared to a
factor of two 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 further
shows 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 variation 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-
13
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 only present 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/dξ)2. The relationship between T and ξ 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 ξ = 0 and 1. This means that
on each side of the stoichiometric mixture fraction, dT/dξ is approximately constant,
and hence the thermal dissipation should be proportional to the mixture fraction
dissipation. Near stoichiometric, dT/dξ 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/dξ near stoichiometric. As a final point regarding
14
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.
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 Ref. [29] it is argued that in nonreacting round jets the mean scalar dissipation
rate should scale as <χ> ∝ (<ξ> / λB) 2, and since <ξ> ∝ x -1 in round jets and
λB ∝ δ Reδ -3/4 ∝ x (since δ ∝ x and Reδ is constant), then we have <χ> ∝ x -4.
Although the measurements are not all consistent, the highly resolved dissipation
data of [22] seem to validate this scaling law in nonreacting 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 radialgradient.
15
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/dξ 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 <χT> ∝ α (<TC> / λB) 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 nonreacting jet, i.e.,
λB ∝ δ Reδ -3/4 ∝ x. If we make the approximation that <TC> ∝ x (which is really
only true for locations upstream of the point of maximum temperature), and we
make the further approximation that α ∝ <TC>1.8 ∝ x1.8, then we find <χT> ∝ x 1.8.
The current data suggest a somewhat weaker dependence on x (<χT> ∝ 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
<TC> ∝ 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 <χT>.
Another possibility is that the assumed state relationship, T = T(ξ), 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
nonpremixed 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
16
temperature gradients are measured by two-point detection, whereas the axial
gradient is measured from the temperature time-series combined with Taylor’s
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
(Reδ ≈ 2000) of the jet flame. 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 to compute the
thermal dissipation. In contrast, nearly lognormal 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 lognormal.
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 dissipation 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 nonreacting jets. A scaling
law is derived for the thermal dissipation rate that accounts for heat release effects
17
and predicts <χT> ∝ x1.8, which 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 spacevarying temperature field of the flame exhibits strong similarities but also clear
differences with the conserved scalar field in nonreacting 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.
ACKNOWLEDGEMENTS
This work was funded by the National Science Foundation under grant CTS9977481.
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19
FIGURE CAPTIONS
Figure 1. Radial profiles of (a) mean and (b) rms temperature at three different axial
stations (x/d=40, 60, 80). Data from TNF workshop data base [14] are
shown for comparison.
Figure 2. Fluctuating temperature power spectra. (a) spectra at the jet flame
centerline (corrected except at x/d = 40 and 50); (b) along ray r/δ=0.4
(corrected except at x/d =40); (c) at x/d=80 and different radial locations,
all corrected.
Figure3. Probability density functions of the normalized radial temperature
gradients along the jet centerline.
Figure 4. Probability density functions of temperature dissipation. (a) log-log plot
showing effect of using radial and axial components to compute
dissipation, (b) variation with radial location at x/d=80, (c) variation with
radial location at x/d=40.
Figure 5. Radial profiles of the normalized mean thermal dissipation (radial
component only). Measurements were made at three downstream
locations: x/d=40, 60, 80.
Figure 6. Variation of the mean thermal dissipation (radial component only) along
the jet flame centerline.
20
2000
Red = 15,200
1200 (a)
800
(a)
-1
10
500
400
χT,r
χT,2D
x/d = 40
x/d = 60
x/d = 80
Lognormal fit
x/d = 40
x/d = 60
x/d = 80
PDF
T (K)
0
10
TNF: x/d = 40
TNF: x/d = 60
TNF: x/d = 80
x/d = 40
x/d = 60
x/d = 80
1600
-2
10
x/d = 40
x/d = 60
x/d = 80
Slope = 1/2
300
200
Trms
400
-3
10
-12
10
(b)
-10
10
-8
0.6
0
5
r/d
10
15
-4
10
-2
10
2
10
χT/(TC/λB)
100
0
-6
10
(b) x/d = 80
r/δ = 0
r/δ = 0.4
r/δ = 0.8
r/δ = 1.0
r/δ = 1.6
20
0.4
Figure 1.
0.2
2
PDF
10
1
10
(a) Centerline
x/d = 40
x/d = 50
x/d = 60
x/d = 70
x/d = 80
0
10
-1
10
0.6
0.0
(c) x/d = 40
0.4
2
10
0.2
-2
1
10
-3
10
2
PSD/(T
- 5/3
(b) r/δ = 0.4
x/d = 40
x/d = 60
x/d = 80
rms
/ fB )
10
2
10
1
10
0
10
10
(c) x/d = 80
r/δ = 0
r/δ = 0.4
r/δ = 0.8
r/δ = 1.0
0
10
-1
10
0.0
-13
10
-1
-11
10
-9
-7
10
10
2
χT/(TC/λB)
-5
10
-2
10
Figure 4.
-3
10
-5
2.0x10
-2
x/d = 40
x/d = 60
x/d = 80
-5
-3
10
-3
-2
10
f/fB
-1
-5
1.0x10
0
10
1.5x10
χT,r / (TC/λB)
2
10
10
-3
10
10
-6
5.0x10
Figure 2.
0.0
1
10
0.0
x/d = 80
x/d = 70
x/d = 60
x/d = 50
x/d = 40
0
Figure 5.
PDF
10
-1
10
-2
10
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1
(∂T/∂r)/(TC/λB)
0.2
0.3
0.4
0.5
Figure 3.
21
0.5
r/δ
1.0
1.5
8
3x10
8
χT,r
2x10
8
1x10
0
30
40
50
60
x/d
70
80
90
Figure 6.
Table 1 Jet flame conditions and estimates of
resolution requirements. (Velocity data from [17],
UC0 = 51.6 m/s is the mean jet exit centerline
velocity which is different from the mean jet exit
velocity U0; temperature data from TNF database
[14]).
λB
x/d
TC
(K)
UC/UC0
δ/d
Reδ
(mm)
fB
(kHz)
40
1580
0.36
5.3
2090
0.32
9.3
50
1745
0.29
6.7
1760
0.45
5.3
60
1840
0.24
8.0
1610
0.58
3.4
70
1680
0.21
9.3
1870
0.60
2.8
80
1470
0.18
10.6
2350
0.58
2.5
22