Turbulent nonpremixed flames of methane near extinction: Probability density functions

COMBUSTION AND FLAME

73: 261-285 (1988)

261

Turbulent Nonpremixed Flames of Methane Near Extinction: Probability Density Functions A. R. MASRI and R. W. BILGER Department of Mechanical Engineering, The University of Sydney, Australia

and R. W. DIBBLE Combustion Research Facility, Sandia National Laboratories, Livermore, CA

Space- and time-resolved measurements of major species concentrations and temperature have been made using the Raman-Rayleigh scattering technique in the blue (visibly soot free) regions of turbulent nonpremixed flames of methane close to extinction. The data are presented in this paper in the form of single variate probability density functions (pdfs) for the mixture fraction, temperature, and the mass fractions of CI-I~,02, H20, H2, CO2, and CO. Representative instantaneous scatter plots and joint lxlfs are also shown. When the mixing rates are low, the data show mostly fully burnt mixtures indicatingthat the chemistry is relatively fast. As the flame approachesextinction, most local mixtures becomeeither partially burnt or simply mixed. The joint pdfs shownbimodality for mixture fractions less than - . 1 and centered distributions for richer mixtures. When close to extinction, fully burnt pockets of fluid are still encountered and these may be responsible for keeping the flame alight. Questions relating to the local, instantaneous flame structure near extinction are discussed in light of existing theoretical models.

1. I N T R O D U C T I O N In a recent paper [1] we have reported the means and rms fluctuations of temperature, density, mixture fraction, and the mass fractions of CI-I4, 02, H20, H2, CO, CO2, and N2 obtained by pulsed spontaneous Raman and Rayleigh scattering in the blue (visibly soot free) regions of a pilot stabilized turbulent jet nonpremixed flame of methane in a coflowing stream of air. In this paper we report information on the probability density functions (pdfs) o f the scalars measured. The burner which produces the flame consists of a central jet of fuel surrounded by a large annulus of co flowing premixed combustion products and this is centered in a much larger coflowing stream This article is in the public domain Published by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, NY 10017

of air. This arrangement has been designed to allow the study of strong turbulence effects on the chemistry under conditions approaching extinction in a flow which is fluid-dynamically simple: a streaming flow describable by parabolic equations. By increasing the fuel jet velocity, the flame approaches extinction which occurs in a region extending from about 15 to 35 fuel jet diameters. At blow-off, the heat release from the pilot annulus maintains combustion of the fuel and outside air over the first 15 diameters. At jet velocities below the extinction condition, the flame develops into a full turbulent jet diffusion flame with a region near x / D j of 20 where mixing is intense (Dj is the fuel jet diameter), temperatures are lowest, and combustion conditions are

0010-2180/88/$0.00

262 most critical. Stab'her and Bilger [2] report initial work on the characterization of the burner with CNG (compressed natural gas, 90.9% CI-I4 by vol.) and LPG 0iquefled petroleum gas 88.0% C3H8 by vol.) fuels. Masri and Bilger [3] report more detailed measurements including thermocoupie measurements of temperatures, LDA measurements of velocity and turbulence, and composition measurements by sample probe. The measurements of these two studies were made on a slightly different version of the burner installed at the University of Sydney, while the measurements reported here and in Masri et al. [1] were made at the Turbulent Diffusion Flame Facility at the Sandia National Laboratories. It is not possible to determine from the probe measurements alone the relative effects of unmixedness and chemical kinetics on the changes in the mean structure of the flame as it approaches extinction (unmixedness [4], describes the effects of the concentration fluctuations on the mean composition; a large standard deviation of the mixture fraction ~7-~1/2 indicates high unmixedness). Space- and time-resolved Raman and Rayleigh measurements can easily address these questions. The means and rms of fluctuations of the Raman and Rayleigh data presented in Masri et al. [1] established that at a fixed axial location in the flame the radial profile of ~7-~1/2 and hence unmixedness remains almost unchanged and the change in the flame structure is due almost entirely to chemical kinetic effects. The probability density functions (pdfs) presented in this paper will help clarify other issues, concerning the turbulencechemistry interactions and the local flame structure near extinction. A large bank of data has been collected at various axial locations in flames ranging from slowly mixing flames to ones close to extinction. At each laser shot the Rayleigh scattering as well as the Raman signals from CH4, 02, I-I20, 1-12,CO, CO2, and N2 are collected simultaneously. The data in its full form consist of 1200-5000 octuplets at each of some 230 measurement conditions. Ideally, when processed, a ten variate lxtf would be obtained for each measured spatial location in the flame (temperature, density, mixture fraction,

A . R . MASRI ET AL. and the mass fractions of seven species). Such multivariate pdfs are of course impractical. In this paper some single and double variate pdfs are of course impractical. In this paper some single and double variate pdfs are presented. More specialized aspects of the interpretation of the data such as in terms of the conditional pdfs or in terms of progress of two or three step reaction models will be reported in forthcoming papers. The processed data will also be made available on computer tapes. Raman measurements in the blue regions of hydrocarbon flames are contaminated by what appears to be "fluorescence" which has an intensity of the same order as the Raman signals. The source molecules of this "fluorescence" are not fully known. Further details can be found in Masri et al. [5]. This difficulty arises only in flames of hydrocarbon fuels, but the problem has been persevered with due to the great economic importance of these fuels. Drake et al. [6] and Dibble et al. [7] have performed Ramaa and Rayleigh measurements in turbulent nonpremixed flames of hydrogen and Drake et al. [8] have recently made Raman measurements in turbulent nonpremixed flames of H2 and CO/H2/N2 at high mixing rates. While the study of extinction in such flames would be of considerable theoretical interest, the kinetics in hydrocarbon flames are fundamentally different due to the role of the fuel in consuming radicals [9, I0], and the mechanism of extinction is thus likely to be completely different. The strategy followed to overcome the "fluorescence" contamination problem is to reduce its effect on the Raman lines as much as possible and then monitor the remaining "fluorescence" (in these experiments the "fluorescence" was monitored at 516.5 nm which is a bandhead of diatomic carbon, C2) and correct for its interference with the Raman lines using "fluorescence" corrections generated from measurements in laminar hydrocarbon flames of known composition. Further details on this correction and the processing of the data can be found in Dibble et al. [11]. This "fluorescence" is detected only in the rich regions of flames (in the mixture fraction range .06 _< ~ _< .25)at temperatures > 1000*C. Therefore, only the data points collected

TURBULENT NONPREMIXED FLAMES at these conditions are expected to show some bias. The nature and magnitude of this bias is discussed in detail in Section 2.2. One of the main objectives of this paper is to present a comprehensive set of data which will assist modelers of turbulent combustion in accounting, correctly, for the chemical kinetic effects which can often be very significant in practical combustors. The bulk of the data is presented in the form of single pdfs for temperature, mixture fraction, and stable species mass fractions. These can be compared to the pdfs calculated directly from full transport equations using the Monte Carlo technique [12, 13]. Otherwise, they assist in the selection of the appropriate shape for the pdfs used in other combustion models [14-18]. Selected joint pdfs as well as examples of scatter plots of the space- and timeresolved data for some measured scalar are also presented. At each position in the flame all of the collected data points are used, without conditioning, in the calculation of the pdfs. The instantaneous local structure of the flame is also discussed. 2. EXPERIMENTAL 2.1. Apparatus and Test Conditions The piloted burner developed at the University of Sydney has been installed in the Sandia Turbulent Diffusion Flame Facility where all the measurements reported here were performed. The burner has a central jet of methane, 7.2 mm in diameter, surrounded by a sto~ichiometric premixed and fully burnt annulus of C2H2, H2, and air with C/H ratio adjusted to be that of methane. The outer diameter of the annulus is 18.0 mm and the lips are thin. The annulus of combustion products from the pilot shields the fuel issuing from the central jet and extends for about 4 fuel jet diameters allowing for any unburnt reactants in the pilot stream to be consumed. During all the measurements the coflow air stream velocity ae and the pilot burnt gas velocity a#, are maintained at 15 and 24 m/s respectively. The pilot burnt gas velocity a # is calculated from known reactant flow rates assuming a product temperature of 2600K. The flame

263 characteristics are then changed by varying the methane bulk jet velocities aj. Four flames have been investigated: Flame

aj (m/s)

Characteristics

A

36

L

41

B

48

M

55

Blue flame up to x~ Oj = 20, electrically connected Blue flame up to x/Dj -- 30, electrically connected Blue flame up to x/Dj = 60, electrically intermittent Almost all blue, electrically intermittent

where x is the distance from the burner's exit plane and Dj is the central jet diameter. Extinction of the flame occurs in the intensely mixed shear layer extending from x/Dj = 15-35, at aj = 65 m/s. The laser beam from the Sandia Combustion Research Facility dye laser (dye: COUMARIN 521, X = 532 nm, AX = 0.3 nm, 0.75 J/pulse, 3 t~s pulse width) is focused to a 500 tim waist diameter. Laser pulse energy is measured with a vacuum- photodiode which received the attenuated laser light before passing through the test section. The Raman scattered light is collected at right angles to the incident beam by a 30 cm focal length, f/2 collection lens and is relayed at 3 x magnification to the entrance slit of a 3/4 m grating spectrometer. The width of the entrance slit (3 mm) determined the length of the Raman probe volume (1 mm), and the height of the probe volume is determined by the laser beam diameter. Ten photomultiplier tubes placed at the exit plane of the polychromator collect the anti-Stokes vibrational Raman scattering from nitrogen, N2(AS); the "fluorescence" intensity monitored at 516.5 nm; the elastically scattered (Rayleigh and possibly Mie) laser light; and the Stokes shifted

The flame is electrically connected when a continuous signal is recorded for the current conducted between a simple wire grid placed at x/Dj = 70 in the flame and the nozzle with a 20 V imposed potential difference. It is electrically intermittent when intermittency spikes start to appear in the recorded current signal.

264

A . R . MASRI ET AL.

vibrational Raman scattering from CO2, 02, CO, N2, CH4, H20 and H2. The "fluorescence" is found to be broadband covering all the visible spectrum. Its effect has been reduced by placing a Polaroid filter at the entrance slit of the spectrometer. The remainder of the "fluorescence" interference on the Raman lines has been corrected using the "flourescence" signal monitored on the C2 line and linear correction coefficients generated using measurements made in laminar diffusion flames of methane. Minimum, mean, and maximum correction coefficients have been derived for each species. In the rich, high t6mperature regions of the flames, when the "fluorescence" correction is high, the Raman signals from CO and CO2 are not too reliable due to the relatively small gain of their phototubes. The nature of this "fluorescence" is discussed in detail in Masri et al. [5]. Further details on the experimental setup, "fluorescence" correction, and the processing of the Raman and Rayleigh signals can be found in Dibble et al. [11].

rich mixtures (.06 _< ~ _< .25) of high temperatures (T > 1000*C). Elsewhere, concentrations and temperature measurements are limited by shot noise and calibration errors and these are less than 5% of maximum values. If very thin reaction fronts exist in the flame, there may be some error introduced by averaging over the probe volume; but it is believed that these effects are not significant. In this section we give estimates of the amount of the data which is affected by substantial "fluorescence" interference. For each of the data points, three "fluorescence" corrections (minimum, mean, and maximum) are applied to each of the Raman lines. When the values resulting from the maximum and minimum "fluorescence" corrections deviate by more than 25 % from the mean "fluorescence" corrected value, then it is considered to have intense "fluorescence" interference. The threshold of 25% is chosen arbitrarily. The values subjected to this test are for the following scalars: temperature T, density p, mixture fraction ~, and the mass fractions of CI-I4, 02, H20, H2, CO, CO2, and N2. Table I shows, for each of these scalars and at each of the axial locations investigated in flames A, L, B, and M, the percentage of data points which are contaminated by intense "fluorescence" interference. These are percent-

2.2 Errors and Biasses

The accuracy of the measurements is dependent on many factors but largely on the severity of the "fluorescence" interference which only exists in

TABLE I

Percentageof Data PointswithIntense "Fluorescence"Interferencein the MixtureFractionRange .06 < ~ _< .25 Threshold

Flame

x/Dj

(%)

Y~2

Yc~

A A A L L L B B .B B M M M M

I0 I0 20 10 20 30 I0 20 30 50 I0 20 30 50

25 I0 25 25 25 25 25 25 25 25 25 25 25 25

5.3 36.3 1.3 0.I 0.0 0,0 3.3 0.0 0.0 0.0 0.5 0.0 0.0 0.0

14.3 ' 48.2 7.8 II.0 3.5 3.7 12.0 0.7 1.7 4.9 8.7 0. I 0.3 3.4

T

~

Yco

Yo2

Yco:

Y,2o

Y,2

P

0.0 2.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

34.9 49.9 13.7 18.8 3.7 3.6 23.9 0.6 0.9 5. I 14.7 0.0 0.0 2.0

73.0 82.2 29.7 35.8 17.5 14.5 67.8 5.2 6.9 15.5 45.2 1.4 1.5 I0.0

65.2 73.9 24.8 31.9 I0.I 9.2 58.1 2.4 2.7 I 1.5 33.8 0.3 0.4 6.2

50.4 58.9 10.5 9.2 3.4 4.4 45.6 2.1 1.9 10.2 20.0 0.6 I.I 5.8

24.4 32.8 3.1 0.3 0.0 0. I 6.8 0.0 0.0 0. I 1,0 0.0 0.0 0.0

51.7 66.3 22.8 22.6 7,7 6,4 37.0 1.8 2.3 7.0 21.7 0.4 0.3 3.5

0.0 3.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

TURBULENT NONPREMIXED FLAMES ages of the data points collected in the mixture fraction range .06 _< ~ ___ .25 at each radial profile. This gives an indication of the error associated with each of these scalars at various measurement conditions. It is worth stressing that such errors are only existent in rich mixtures (.06 < ~ ___ .25) at high temperatures ( T > 1000*C). It is seen that flame A atx/Dj = 10 has the highest "fluorescence" emission and that the interference is highest on the CO and 02 lines. The fact that temperature and density show no "fluorescence" contamination is somewhat misleading because the 25 % threshold is high. Table I shows for flame A at x/Dj = 10 that if the threshold is lowered to 10%, the percentage of data points with intense "fluorescence" interference on temperature and density increases to 2.1% and 3.9%, respectively. In the instantaneous scatter plots of Section 3.1 (Figs. 1 and 2) the points with intense "fluorescence" interference are excluded. This leads to a low density of points in the region .06 ___ ~ _< .25 at x/Dj = 10 and 50 where the "fluorescence" is intense. In the calculations of the pdfs presented in Sections 3.2 and 3.3, all the data points are used including the ones with intense "fluorescence" interference. This leads to values for the pdfs of all the scalars slightly biased toward higher values in the regions of intense "fluorescence." The effects of the uncertainty coming from this interference on the mean values and standard deviations are presented in Masri et al. [1].

3. RESULTS AND DISCUSSION 3.1. Scatter Plots Examples of the instantaneous data are presented in Figs. 1 and 2. Each plot includes the data collected over the whole radial profile excluding the points with intense "fluorescence" interference. About 13,000-28,000 data points are shown on each plot depending on the axial location. Such plots are useful in showing the qualitative features of the data. True joint pdfs at a single spatial point are presented in Section 3.3. Figure 1 shows four scatter plots for the instantaneous number density of CI-h versus that

265 of 02 for flame B at x/Dj = 10, 20, 30, and 50. The number density is obtained directly from the measured Raman signals multiplied by the corresponding calibration factor. The solid and broken lines drawn on Fig. 1 are lines of constant mixture fraction ~ and reactedness b, respectively. The reactedness b at a certain value of mixture fraction is determined as follows:

b

=

Pi--Pm i ' Ps,i-lam,i '

(1)

where Pi is a scalar representing temperature or species mass fractions, Pm,i is the frozen value of the scalar i (Pm,i = 0 for H20 and CO2 and Pm,i = 300 for temperature), and Ps,i is the value obtained from the predictions of the flame sheet model [19]. The values of b vary between zero for frozen and one for fully burnt mixtures. Almost all the data points at x/Dj = 10 are for mixtures more than 50% reacted implying that the chemistry is fast. The chemical kinetic effects which normally arise near the nozzle exit due to sharp concentration gradients are apparently overwhelmed by heat already released from the pilot flame. At x/Dj = 20 and 30, the reactedness decreases sharply and a large number of data points on both sides of stoichiometry are for mixtures less than 30% reacted (the stoichiometric mixture fraction for CH4 is ~s = .055). The mixing is very intense in this region and the chemical kinetics are not fast enough to consume the fuel and oxygen which coexist in large concentrations. These effects gradually decrease further downstream as the intensity of the mixing is relaxed, but still persist down to x/Dj = 50 where a percentage of the data have a reactedness between. 1 and .3. Figure 2 shows four scatter plots for temperature versus mixture fraction for flame B at x/Dj = 10, 20, 30, and 50. The dotted and solid lines represent the predictions of Miller et al. [20] for a counterflow laminar diffusion flame of methane with a = 1 and 320 s- 1, respectively, where a is a stretch parameter indicating how close the flame is to extinction (which occurs at a = 340 s-~). These represent the theoretical limits for laminar diffusion flamelets. The frozen chemistry values and the predictions for the a = 1 s-l flame form

266

A.R. MASRI ET AL FLAil[" (,~

FLAUE ( ~

u/= 48 m~f'

X,/D~= 10

Uj= 48 ms-1

X/t), = 20

,? v

o

,o

20 [N]o~

~o 40 .~o,~(~ )

FLAI,IE ~ )

50

T.o

2.0

Uj= 48 ms-'

3.0

,o

2.0

3.0

[N]o ,

4..0

.10-" ( ~ "~ )

FLAME ~ )

uj= 48 m6-1

X/O, = 50

X/D, = 30

o

o

4.o

5.o

Fig. I. Scatter plots of the instantaneous number dcnsifins of CH4 versus O2 at four axial locations in flame B. The solid lines are for constant mixture fraction ~ and the dashed lines for constant rcactcdness b as predicted by the flame sheet model (Rcf. [19]).

5,0

TURBULENT NONPREMIXED FLAMES

FLAME ( ~

267

"U~=48 m~~

FLAL~E ~')

X,/Di = I0 I

'

I

'

Uj= 48 ms-'

X/1Dj= 20 I

. .,.."

7 ,.

"...'~...

?i:::I.. > "

i . .. ::;.:'::.:

,.

.~

..'!:.~ :~!

• ..

500

0.05 0.10 0.3 blIXI'UREFRACT ION

0.5

0.7

0

FLA"E (8~ U~: 48 ms-'

FLAWIE (~

×/l)j= 30

0.05 0.10 0.3 MIXTURE FRACTION ,

0.05 0.10 0.3 MIXTURE FRACT ION

0..5

0.7

0.5

0.7

Uj= 48 ms-'

X/Dj = 50

0.5

0.7

0

0.05 0.10 0.3 MIXTURE FRACTION

Fig. 2. Scatter plots o f the instantaneous temperature T versus ndxtu~ fraction ~ at four axial locations in flame B. The dashed and solid lines are for the a = I and 320 s-J flames, respectively, as predicted in Ref. [20].

268

A. R. MASRI ET AL.

the theoretical bounds within which the data should lie. Temperature is obtained from Rayleigh

scattering and the mixture fraction ~ is calculated from the H atom balance as follows:

= [((4.032/16.032)Ycu4 + (2.016/18.016) YH20 + YH2) -- YH,e] YH,j-- Yn,e where Yzis the mass fraction of species i, YHj and YH.e are the mass fractions of hydrogen atoms in the main fuel and air, respectively, and 2"016(~+1 ) 18.016 where oz is the specific humidity (kg H20/kg dry air) in the inlet air. The humidity was frequently measured using a hygrometer. Almost all the data points at x/Dj = 10 lie between or close to the laminar flame predictions for a = 1 and 320 s -~. The "fluorescence" interference is responsible for the' data points with temperatures higher than the theoretical limit at a = 1 s- 1. At x/Dj = 20 and 30 most of the data points are scattered below the theoretical limit of the a = 320 s-1 flame. This indicates that the mixtures are either partially burnt or fully unburnt. The departure of the unburnt mixtures from the mixing temperature of 300K is possibly due to mixing with fully burnt pilot gases. The percentage of fully burnt mixtures increases and most of the flame is ignited at x/Dj = 50 as the mixing intensity decreases. Although the data are scattered right across the temperature range, it shows some tendency to bimodality (i.e., either fully burnt or unburnt mixtures are encountered) for lean mixtures.

3.2. Single pdfs A comprehensive set of data is presented in Figs. 3-14 in the form of single variate pdfs of mixture fraction, temperature, and species mass fractions. The temperature is obtained from Rayleigh scattering and the mixture fracture is calculated as in Eq. (1). For each location in the flame,1250 instantaneous and spatially resolved data points have been collected all of which are used in the calculation of the single pdfs including the ones

(2)

with intense "fluorescence" effects. The area under each pdf curve equals one. On each figure, four diagrams are presented each of which refers to a certain axial location in the flame. For clarity, only four single pdfs are shown for each radial profile. On each plot the radial location and the relevant means and rms of fluctuations are al~o presented. Favre averages are calculated for mixture fraction and species mass fractions and conventional averages for temperature and intermittency. The intermittency factor 3' is a measure of frequency or fraction of time the measuring volume is occupied by turbulent fluid contaminated by material originating from the jet (3' = 0 when the measuring volume is always in nonturbulent fluid and 3' = 1 when in turbulent fluid). Two different thresholds of the mixture fraction, ~th = .0008 and ~m = .0012, are used to discriminate between the turbulent and the nonturbulent zones or between jet-mixed fluid and air (i.e., the flow is taken as turbulent when ~ _> ~th and nonturbulent with ~ < ~th). These thresholds are determined from measurements made in the air stream and processed like the rest of the data. It is found that about 2 % of the air data have values of mixture fraction, ~ > .0012 and about 10% have values of _> .0008. For each of these arbitrarily selected threshold values, ~th = .0008 and ~th = .0012, the conventional intermittency factor 3' is calculated and labeled (A) and (B), respectively. Although this method gives adequate values for the intermittency factor, a more accurate method is that used by Bilger et al. [21] for calculating 3' using a tedious least squares fit of a Gaussian curve to the pdfs of temperature measured in heated wakes, Figures 3 to 14 show the single lxlfs of mixture fraction, temperature, and the mass fractions of CI-Lt, 02, H20, 1-I2, CO2, and CO at various locations in flames A, L, B, and M. Due to space limitations, data at selected axial locations in the

0.0

OkX_ . , 0.0 O.L

•

0.1

A

i

0.2

I

0.4

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l

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~

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•

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20

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20

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0,1

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0,t8

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FAVRE MEAN Rl,f~

I 0.6

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r
,

0.

F" (~)

= lO a n d 50.

MIXTURE FRACTION

,.~.~,u[,mar. , 0.2 v..1.4

Oj= 55 m~~

MIXTURE FRACTION FLAME ( ~

0 i / x..,~

o.. LB

~v

30

40

Oj= 48 m~!

X/Oj = 50

FLAME (~)

Fig. 3. Single pdfs o f the m i x t u r e fraction ~ in flames B a n d M at

,

016

r" <.)

.

0.6

I~. 4

r" <--) ME~F^VRFI~Ms

MIXTURE FRACTION

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Oj: 55 m~'

MIXTURE ~RACT~ON

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"t

30

40

FLAME ( ~

2_

r.Lg

LO

20

30

40

~.~,_

r,/'~.15.Z~

B:87

i

r.lkg

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20

30

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r.tS.Z~

CO,IV. INT. (^)
1.0

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COl,IV. INT. (A) (B)

h,,} (3",

>

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= 20 in flames A, L, B, and M.

8. I

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x/Dj

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46

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38

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J

CONV.(^,~I~).

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r (,,,,) ~^V1~j~

g. 2 0. 4 8.'6 MIXTURE FRACTION P+

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i

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1

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i

XlOj =

FLAME ~

•

~ . .

Oj - 315 .+'

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l.{}

1.0 C(A[~). {~)T.

p.&g

16

36

46

t6

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38

46

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>

>

>

t~

TURBULENT

NONPREMIXED

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271

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TEMPERATURE .

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1400

TEMPERATURE , FLAME M~

600

FLAME

(--)

(K)

-

7. !

0. 8

r

(K)

1800

r (--)

Fig: 6. Single pdfs of temperature T at x / D j = 20 in flames A, L, B, and M.

T

1400

TEMPERATURE .

I

TEMPERATURE .

B~

Oj- 30 =6'

X/Oj= 28

FLAME ~

RMS

2200

--

414

g5

CONVENr 628

MEAN

2200

C0~EN MEAN RMrS

r~[12.

7~

6

g

12

5

g

12

>

nl

>

36

40

C~

g.O

,U.

0. 4

0.~6

Oj= 36 .,;;

0.~8

1,1.2

r" (--)

r w)

0,2

Z..

MEANFA~V~

z,.

LO

20

36

40

LO

20

30

40

0-

/

30

40

0.0

LO

0,

o

,o

v20

O-

v 20

30

40

Fig. 7. Single pdfs o f the m a s s fraction o f fuel Ycr~ at

0.4 0.6 6.8 1.9 MASS FRACTION OF CH4 . YCH4

. . . . . . . . .

Oj= 48 ~ '

I. 0

I /4.2~

~.,,.,.

MEANFAVRMR~

MASS FRACT[ON OF CH 4 , YCH4

0. 2

FLAME (~)

It

0 0.0

I0

v~O

5

CL

v 20

30

40

FLAME (~) =

,

I

20

I

,

I

"

Oj= 41 IB61

i

~

I

16.9

7. ! !z.z

~.~

~

,

x/Dj =

'

r" <--)

0.'4 g.'B o.'e

Oj= 55 ,,,dr

~

I/r.l~lJ--

"'

2 0 in flames A , L, B, a n d M.

'

I.'O~/~''L'-

MEAN F^~

MASS FRACTION OF CH4 , YCH4

0.'2

FLAME (~)

~.,~

8i~

~.

F^VRE i~ (m~) MEAN RMS

0.2 0.4 0.6 0.8 L,O MASS FRACTION OF CH4 ,YcH4

I

X/Dj

FLAME ~

7

r.o.i

10

20

30

46

I r'Le

LO

20

30

40

--

>

X

0 Z

Z

Z

v

88 r

28

i ~

'

~

Oj- S5 .~'

~

r <--) ~

F^}~ 88

' ........

a

2@

61}

8@

~.88

8.'L:~ '

r (--)

Oj= 55 m~'

r <--)

8.85 RI8 8.15 8.28 gASS FRACTIONOF 02 . Y0~

FLAHE (E)

MASS FRACT[ONOF 02 , YD~

8. L5

,X_.+,

Oj= 48 .~1

8.85 ' 8.10

_J.J,

FLAME (~)

8.25

MEANFA~j

8.25

14E:AN F^~I~

Fig. 8. Single pdfs of the mass fraction of oxygen Yo2 in flames B and M at x/Dj = lO and 50,

MASS FRACTIONOF O~ . YO~

FLAME ~

8.88"

8 / k

2B

6B •

28

8.J25 I -

4{}

88

:_. 4e

, , , , 8 . 8 5 ~.I18 8 J 1 5 8.28 MASSFRACTIONOF 0z . Yoz

IlL4

~+

r (..)

48

..~,68

O_

N

8e

Oj - 48 m~'

x/o+-le

FLAHE ( ~

•

20

40

60

180

ro.~+.. 3~

\°

48

6g

B~

>

~n

>

>

4~

T U R B U L E N T N O N P R E M I X E D FLAMES

275

!

,3

4 W 7*lD E

<

1~" <~

"

|

t

i

I

I

(zO~) d

(z°), ) d

I

I

0

m

c~ t

g

#

J i

(Zo~) d

I

I

(zn).) d

|

.~

v

n

v

g. ee

g. O8

ME~IV~Rlul5

ILl6

I r ~ - 7. i m

PiL

50

lee

/ ~ r . 6. 4 m

I

P.Lt

5B

lee

15B

l

>-

.

B B. ee

/11

I'x

~

.

, "~.

g. 08

x

,.

11.12

I

8. lG

I

r (--) ~ ^ ~ g ~ S

MASS FRACTION OF HzO .YHzo

0.114

,

Oj- 55 rod'

B. 15

I

~,!, t| t!~

. ~,,.,) I ~ ^ v ~

0. B4 B. 08 0.12 MASSFRACTIONOF HzO .YH20

FLAME(~

B. ee

158 F I

lee

150

Oj= 41 I.I/l

x,°.~,

FLAME (~

Fig. 10. Single lxlfs of the mass fraction of water YH2Oat x/Dj = 20 in flames A, L, B, and M.

,'i

i

,./.,~,_

~"+ tli tit

0.12 MASSFRACT[ON06" HzO .YHz0

B.04

x,o.+

r'~..~

g.lO8 0.i12 MASSFRACTIONOF H20 ,¥H20

FLAME l~) Oj= 48 .171

, Y

50

lee

150

g. ee

I

0. 10

/ ..¢ -'~,.___

~'+ tli f!ti

x,o,..

.J .~-._._ 0.04

-

-

lee

50

-

150

r" (,,.,,o ~F^VRF/t~

FLAME (~) Oj : 36 ,i.i'

i

. . u -

lee

150

5g

lee

150

m

~o

hJ "..d

I

2

3

4

2

3

4

5

I,!, II III

r(_, .E,~^~,~

5

IIg

"

>~

*

,? 4

Fig. 11. S i n g l e pdfs o f the mass f ~ c d o n 50.

MASS FRACTION OF H z . Y H z * I 8 s

6

~

~

Oa= 55

2

3

ms t

4

x/Dj

= ] 0 and

~ ~.,-

r'(,,,,,) I~ANFAVR£RMS

MASS FRACTION OF H z . YHz*L6 a

I

. . . . . . .

FLAME ( ~ )

I

im

i

r•.z4,

~"-

IM ID

r(--) ~^VRERHs

MASSFRACTIONOF Ha . YHz*103

<.~ ~

Oj= 48 =6t

x,0,..

FLAME (~)

o f h y d r o g e n Yx2 in f l a m e s B and M at

v

I

%- SS. '

x,0,.,,

FL^ME®

MASSFRACTIONOF Hz ,YH~,'IBa

Z...

2

4

o_ 2

6

.

r/.T, 9P'4. II

*

>v

?

o_ 2

4

9

. ~ , . , . , , , ,

,-<-, ,',E..~^~,~ !~, f B IE~

v

*

'7

v o_

'7,

~* ~8 .' x%= ,.

FLAME®

r.lLI

2

4

".Lil

-,..I

m

t-'

0 Z

,.-]

m :Z

L-'

,--t

278

A . R . MASRI ET AL

,=_~

-~

t-gI"

~:~

Jl i I 1",~ ~"

( X~, ) d

'

EO[*

'

(ZX~,) d

'

~'

-r,k,,

,. I

I

¢.i11-

I

(ZX),) d

I

,\.. i

¢_61.

.~

(ZX,~) d

0.m

.

0.12

,

-

,i 0.08

,

oj - 4e ,,,;'

.-

n. IZ

,

r" ~,~

|/ltlm

I

r.L|

58

L08

158

,~.+.o _

r.LI

58

tm

158

0.08

8

v108

(3.

^158

288

0.08

B

58

~15~

0.08

oj . 41 . '

0.12

~ ~-,

0.08

0.112

e

?; f|

MASS FRACTION OF COz . YCO=

0.84

x~

®

0.16

I

0. L6

.~^'%~

MASS FRACTION OF COz . YCOz

0.04

FuAMe©

Fig. 13. Single pdfs of the mass fraction of carbon dioxide Yco2 at x/Dj = 20 in flames A, L, B, and M.

11.16

w.~^~

0.L6

i/4,2m

l

tl Iil

r~ (-,,) ]~-.~F^VRFI~is

MASS FRACTION OF COz . Ycoz

0.04

,

X/[

FLtJ4E ( ~

B a'+'~'-.- ~ _

~158

288

, 0.08

Oj= 36 m;'

MASS FRACTION OF COz . YCOz

Ok/".-..._ ..... 0.08 0.04

^158

FLAME ( ~

i

r=7.1

r.ILl

58

lW

150

m

ro &| m

58

l~

158

I'..) "-..I

nl

Z 0 Z

L" m :Z ,...]

A

6

1Be

38e

4m

/

OO.6e

1Be

O..

o

4ee

Oj- 49 ,,,roT'

,

~!~

, YCO

.Om

26e

38e

466

..

46O

m

o,. ,l . '

i

,. [[ [t~

,.-<,,,,, .E~"%~

6.68

4o9

i =

. YCO

20 in flames A,

14A~ FRACTION OF CO

,

/

i

I P.~t.t¢ elm

,

[|

0.10

lii., It

8.192 ' PI.~Pl4 ' e.lO6 ' 0.i88

x,o~.~

OO.OO" O.IO2 ' 6.'04 ' 9.106 9.'88 ' 9.'10 ~''IUMASS FRACTION OF CO , YCO

3OO

4W

FLA~ ©

Fig. 14. Single pdfs of the mass fraction of carbon monoxide Yco at x / D L, B, M.

MASS FRACTION OF CO

I

![ t~i

.

i P ~ - i¢ Zm

r (--) XF.A~^VRF/~S

,

,--<_> . ~ " " ~

O.IO2 ' B.~64 ' 0,'86 ' 0.'88 " 8. LO

x,o~.~,

FLAME (~)

i

%- ~ ~ ,

9.102 0.J64 0.~96 0.'68 9. 19 HASS FRACTION OF CO • YCO

FL^WE ®

~7.1

r]....

IW

m

r.Ll

Im

39O

400

m

I'-"',

>

>

>

i,,J oo

TURBULENT NONPREMIXED FLAMES flames will only be presented. Figures 3 and 4 present the pdfs of the mixture fraction ~ with Fig. 3 showing those for flames B and M at x/Dj = 10 and 50, and Fig. 4 showing the pdfs at x/Dj = 20 in flames A, L, B, and M. Two values for the conventional intermittency factor 3, are shown for each radial location. These have been obtained using threshold values of .0008 and .0012 and are labeled (A) and (B), respectively. Near the outer edges of the flame, the values of 3' are obviously sensitive to the value of ~th- A near Gaussian distribution prevails for the pdfs of ~ on the flame centerline at r = 0 except near the nozzle at x/Dj = 10 in flame B (r is the radial distance from the flame centerline). The distribution broadens, coveting a large range of ~, as the rich side of the visible flame brush is approached. Within the turbulent fluid, the distribution is continuous and there are no sharp changes in the shape of the pdf of ~ except near the flame edges where pockets of nonturbulent fresh air are encountered and intermittency spikes start to develop. These results are consistent with other pdfs of conserved scalars measured in cold jets [22, 23], heated jets [24, 25], and jet diffusion flames of hydrogen [26-28]. At a fixed axial location in flames A, L, B, and M, the shapes of the pdfs of ~ remain essentially unchanged with increasing mean jet velocity, aj. The mixing characteristics determined by ~ and its rms of fluctuations [1] are therefore largely independent of the chemical kinetic effects associated with the increase in aj. Various distributions have been used to describe the pdf of the conserved scalar and these are disc~assed in detail by Bilger [29]. As it can be seen from the pdfs of ~, the separation between turbulent and nonturbulent fluid is not well defined due to what Corrsin [30] called the viscous superlayer. Effelsberg and Peters [31] have developed a composite model to account for the effects of this superlayer on the shape of the pdf of ~. The pdfs of temperature are presented in Figs. 5 and 6 for the same axial locations in the flames as those shown for the pdfs of ~ in Figs. 3 and 4. The pdfs of temperature have a Gaussian distribution on the flame centerline and an intermittency spike on the air side. This is qualitatively similar to previously reported pdfs of T measured in turbu-

281 lent diffusion flames of hydrogen [6, 7] and methane [32]. In regions of the flame where the mixing rates are low as at x/Dj = 50 in flames B and M, the pdfs of T are bimodal at r = 24.1 and 26.7 mm, respectively. This is qualitatively similar to the pdfs of T that would be obtained from fast chemistry [29] using the pdfs of ~ shown in Figs. 3 and 4. This is not the case, however, in the intense mixing regions at x/Dj = 20 where the pelfs of T, at r = 11.9 mm in flame B and at r = 12.7 mm in flame M, lose their bimodality and are mainly confined to temperatures less than 1000K. This indicates that the chemical kinetic effects are strong in these regions of the flames and that the mixtures are not fully reacted. In regions of the flame where the mixture fraction lies within the reaction limits, the high temperature tail ( T > 1000 K) of the pdfs of T indicates the existence of hot burnt pockets which may be responsible for keeping the flame alight (the reaction limits are determined by the region of high reaction rates which, according to the calculations of Miller et al. [20] for a methane diffusion flame, is independent of stretch and lies between ~ = .028 and ~ = .087). The frequency of encountering these burnt pockets decreases sharply as the flame approaches extinction. This is clearly observed from the pdfs of T at x/Dj = 20 as we progress from flame A (r = 10.6 mm) to L (r = l l . 2 m m ) t o B ( r = 11.9 mm) to M (r = 12.7 mm). This effect is also associated with a decrease in the means and the rms of fluctuations of temperature presented in Masri et al. [1]. The pdfs of density p are not shown due to space limitations but are consistent with the pdfs of T. They are near Gaussian except in the reaction zones where they become bimodal in regions of low mixing rates as at x/Dj = 50 in flames B and M or distributed if the chemical kinetic effects are substantial as at x/Dj = 20 in the same flames. The pdfs of the mass fractions of CH4 and O2 at x/Dj = 20 in flames A, L, B, and M are shown in Figs. 7 and 8, respectively. It should be noted that for clarity, the pdfs of the mass fraction of 02, Yo:, are plotted in the reverse radial order to those of the mass fraction of CI-I4, YcH4. Considering the same radial location a t x/Dj = 2 0 , the mass fractions of both CI-h and 02 increase as the

282 intensity of the mixing rates increase from flames A to M and the penetration of 02 persists to reach the centerline of the flame. As the mixing rates become more intense, as at x/Dj = 20 in flames B (r = 11.9 ram) and M (r = 12.7 ram), the pdfs of Ycx4 become similar to the mixing pdfs of ~ shown in Figs. 3 and 4 and the pdfs of Yo2 become broad and distributed over a large range of oxygen mass fractious indicating partial reactedness. Figure 9 shows the pdfs of Yo2 at x/Dj = 10 and 50 in flames B and M. Hot combustion products from the pilot are responsible for keeping the flame alight.at x/Dj -- 10. At low mixing rares, such as at x/Dj = 50 in flames B and M at r = 24.1 and 26.7 ram, respectively, the pdfs of Yo2 are bimodal as expected from the bivariate behavior of reactedness. Figure lO shows the pdfs of the mass fractions of H20 at x/Dj = 20 in flames A, L, B, and M. With increasing jet velocity from flames A to M, the pdfs are shifted toward lower values of YH20 indicating lower reactedness. The pdfs of YH2Oare consistent with those of temperature and show that at x/Dj = 20 in flame M, which is close to extinction, fully burnt pockets or streaks of fluids are still encountered at r = 12.7 ram suggesting that these may be responsible for keeping the flame alight. The pdfs of Ya2 are presented in Fig. 11 for flames B and M at x/Dj = 10 and 50 and in Fig. 12 for flames A, L, B, and M at x/Dj = 20. At x~ Dj = 50, the pdfs of Ya2 are slightly himodal at r = ~5.2 mm in both flames B and M. This bim0dality is also expected from the predictions of Miller et al. [20] for the a = 1 s-l methane diffusion flame using the pdfs of ~ shown in Fig. 3. In regions of strong chemical kinetic effects such as at x/Dj -- 20 in flames B and M, the pdfs of YH2 regain their Gaussian like distributions and peak at low mass fractions of H2 indicating partial reactions and little formation of combustion prodUCts: Figures 13 and 14 show the pdfs of CO2 and CO, respectively, atx/Dj -- 20 in flames A, L, B, and M. The pdfs of Yco2 and Yco behave similarly to those of YH20 and YH2, respectively. The bimodality of the pdfs of Yc02 and sometimes Yco, in the rich regions of the flames where the mixing

A . R . MASRI ET AL. intensity is low as at x/Dj = 10, is misleading as this is only due to the error arising from the "fluorescence" correction which can exceed the magnitudes of the CO2 and CO Raman signals on the rich side of slowly mixing flames. The pdfs of the mass fractions of N2 are not presented due to space limitations but they behave similarly to the pdfs of the conserved scalar, ~. Unlike the single pdfs of the conserved scalars, the pdfs of the reactive scalars are very sensitive to the chemical kinetics. In regions of low mixing rates they show the himodality expected for fast chemistry from the pdfs of ~. When the mixing rates intensify and the chemical kinetic effects become significant, the bimodality is lost, mixtures are partially reacted, less combustion products are formed, and reactants coexist in relatively large concentrations. The pdfs of all reactive scalars are consistent with one another and lie within the limits expected for the fast chemistry and frozen chemistry from the Ixlfs of ~. The pdfs of the product species taken in regions where the mixture fractions lie within the reaction limits show a long tail toward fully reacted values for T, YH2O, and Yco2. These tails get thinner as mixing becomes more intense and the flames approach extinction. This suggests that hot streaks or fully burnt pockets of fluid are involved in keeping the flame alight and that the frequency of their occurrence decreases as the flame approaches extinction.

3.3. Joint pdfs Various locations were selected in flames L, B, and M and about 5000 instantaneous data points were collected at each location. These data have been used to calculate a number of joint probability density functions, jpdfs. Examples of these jpdfs are presented in Figs. 15-18. Each figure shows four jpdfs each of which is labeled and refers to a different location or a different flame. The labels are named as follows: JXXFRRnn where XX is the axial location (x/Dj), F is the flame code, RR refers to the radial location, and nn is a file number for the jlxlf plotted. Details of the four jlxlfs covered in each figure are given

TURBULENT NONPREMIXED FLAMES below: Label J20Lllnn J20L15nn J20Mllnn J2OM15nn

Flame

x/Dj

r/R

(

77"(K)

L L M M

20 20 20 20

2.47 3.32 2.47 3.32

.239 .037 .229 .068

1208 1000 786 647

Here ~-is the Favre averaged mixture fraction, f'is the conventionally averaged temperature, and R is the fuel jet radius (3.6 mm). On each plot, a total of 19 colors are shown representing 19 levels for the probability density. The split between levels is near logarithmic and the value of the probability density at each color level change is shown next to the color bar. The volume under each jpdf surface equals one. Figures 15-17 show the jpdfs for temperature T versus ~, the mass fraction of water YH20versus ~, and Yo2 versus ~. All jpdfs are found to lie in the domain expected from fast and frozen chemistry for the pdfs of ~ except for few overshoots in temperature and species mass fractions detected in the regions of intense "fluorescence." For mixture fractions less than = . 1 the jpdf taken at r/R = 3.32 in flames L and M shows a distinctly bimodal behavior with T, Ya2o, and Yo2 following either a fully burnt line or a line representing only mixing with hot combustion products. This bimodality disappears in these figures for richer mixtures, the reactedness at any ~ becoming centrally distributed. This is even more evident for the jpdfs taken at r/R = 2.47. A similar effect is noted in Fig. 18 which shows jpdfs of the mass fraction of fuel versus that of oxygen. The bimodal behavior for ~ less than - . 1 is transformed into a partially reacted or partially premixed centered distribution on the richer side. Thejpdf of (Ycn4, Yo2) atx/Dj = 20 in flame M shows that most samples encountered are either fully mixed or only partially reacted and fuel and oxygen coexist in large mass fractions. The jpdfs of all reactive scalars, including the ones not presented in this paper, are consistent with one another and show that in regions of low mixing rates most of the data collected lie close to

283 the fast chemistry limit indicating complete combustion. When the mixing rates become intense, at x/Dj = 20 in flame M, the majority of the samples are either partially reacted or mixed with hot combustion products and lie on an asymptote slightly shifted from the cold mixing one. A number of hot fully burnt pockets are still encountered, however, and the probability densities of such pockets are about an order of magnitude lower than those of flame L at the same location. The existence of such hot streaks in regions of the flames which are very close to extinction indicates that they may be responsible for keeping the flame alight and that full flame blow-off will therefore take place when these hot streaks cease to occur or when their probability density becomes small. The cross correlations between the various stable species give little information on the kinetics scheme which adequately represents the chemistry of these flames. Such information could only be obtained from conditional sampling of the data with respect to mixture fraction. Such analysis will appear in forthcoming papers. 3.4. Inferences on Flame Structure

In turbulent reacting fows, the understanding of the local, instantaneous structure of the flame as it approaches extinction is essential to the development of theoretical models which account for finite rate chemistry effects and predict overall flame extinction. In the laminar flamelet model [15, 16, 33], the turbulent flame is conceived as an ensemble of moving laminar diffusion flamelets which can be extinguished as the local mixing rates become intense. Overall flame extinction occurs when the percentage of the extinguished flamelets exceeds a certain limit. At any instant of time the local mixtures can be either burnt or unburnt. This is consistent with the bimodal behavior seen in Figs. 15-18 for ~ less than - . 1 . But the bimodality alone does not provide conclusive evidence that laminar flamelets prevail because well stirred reactor data would give a similar behavior for mixtures within the reaction limits (.028 _< ~ _< .087 for Cl-I4 diffusion flames which are close to the flammability limits for premixed CH4-air mixtures). The data show, however, that

284 the bimodality extends beyond the reaction limits. This supports the existence of laminar flamelets which are likely to be in the superlayer region which extends to mixture fractions greater than .2 [22]. The existence of folded flamelets also cannot be excluded. For richer mixtures (~ > . 1), the data show a centered distribution with a high probability of intermediate states between burnt and unburnt flamelets. Such mixtures can represent any of three possibilities (i) nonstationary transitions between the fully burnt and extinguished flamelets (i.e., mixtures lying on the unsteady transition, middle branch of the S shaped flame stability curve); (ii) locally, partially premixed flamelets which were proposed by Peters [34] as an extension to the laminar flamelet model and later adopted by Rogg et al. [35] with detailed chemistry and using various degrees of partial premixing to predict methane-air diffusion flames; (iii) partially reacted mixtures of fuel, oxidant, and product which lie beyond the rich reaction limit, have low rates of reaction, and yield therefore a broad, distributed reaction zone. The existence of such distributed reaction zones in turbulent reacting flows require that their thickness be larger than the smallest representative turbulence length scale [36]. These questions are to be addressed more quantitatively in forthcoming papers.

A . R . MASRI ET AL. iii. The pdfs of temperature and product species, taken in regions where the mixture fractions lie within the reaction limits, indicate that, near extinction, fully burnt pockets of fluid are still encountered and may be responsible for keeping the flame alight. iv. The jpdfs are found to lie substantially within the expected domain and are consistent with one another. They show bimodality for mixtures with ~ less than - . 1 and a centered distribution for richer mixtures. v. Although the bimodality detected on the lean side favors the thin laminar flamelet model, distributed reaction zones or even folded flamelets may also be possible. The centered distribution implies either unsteady transitions between burnt and unburnt states or partially premixed flamelets or broad reaction zones with low reactedness. These points need further elucidation using conditionally sampled pdfs.

This work has been supported by the U.S. Department o f Energy, Office o f Basic Energy Sciences, Division of Chemical Sciences, the Australian Research Grants Scheme, and the Garrett Turbine Engine Company o f Phoenix, Arizona. REFERENCES

4. CONCLUSIONS i. The pdfs of the conserved scalars like mixture fraction ~ and the mass fraction of nitrogen YN2 are qualitatively similar to those reported in nonreacting jets and reacting jets with fast chemistry. Their shapes do not depend on the mean jet velocity aj and hence are substantially independent of the chemical kinetic effects. ii. The pdfs of the reactive scalars, like mass fractions of the reactants and products, are very sensitive to the chemical kinetics which cause an overall increase in reactants and decrease in combustion products. They are consistent with one another and lie within the bounds expected for fast chemistry and frozen chemistry from the mixture fraction pdfs.

1. Masri, A. R., Dibble, R. W., and Bilger, R. W., Combust. Flame 71:245-266 (1988). 2. Sdrner, S. H., and Bilger, R. W., Combust. Flame 61:29-38 (1985). 3. Masri, A. R., and Bilger, R. W., Twenty-First Symposium (International) in Combustion, The Combustion Institute, Pittsburgh (1988) pp 1511-1520. 4. Hawthorne, W. R., WeddeU, D. S., and Hottel, H. C., Third Symposium on Combustion, Flame and Explosion Phenomena, Williams and Wilkins, Baltimore, 1949, p. 267. 5. Masri, A. R., Bilger, R. W., and Dibble, R. W., Combust. Flame 68:109-119 (1987). 6. Drake, M. C., Pitz, R. W., and Lapp, M., 22nd Aerospace Sciences Meeting, Paper AIAA-84-0544, Jail. 1984. 7. Dibble, R. W., Kollmann, W., and Scheffer, R. W., Combust. Flame 55:307-321 (1984). 8. Drake, M. C., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh (1988) pp 1579-1589.

TURBULENT NONPREMIXED 9.

10.

11. 12.

13. 14. 15, 16, 17, 18.

19. 20.

21. 22. 23.

FLAMES

Bilger, R. W., and Kee, R. J., "Simplified Kinetics for Diffusion Flames of Methane in Air," Proceedings of Joint Conference Western States Section and Japan Sections, the Combustion Institute, Hawaii, November 1987, pp. 277-279. Peters, N., in Numerical Simulation of Combustion Phenomena (R. Glowinski, B. Larrouturou, and R. Temam, Eds.), Lecture Notes in Physics, Vol. 241, Springer, 1985, p. 90. Dibble, R. W., Masri, A. R., and Bilger, R. W., Combust. Flame 67:189-206 (1987). Pope, S. B., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1985, p. 1001. Pope, S. B., Prog. Energy Combust, Sci. 11:119-192 (1985). Borghi, R., and Gonzalez, M., Combust. Flame 63:239-250 (1986). Peters, N., Prog. Energy Combust. Sci. 10:319-339 (1984). Liew, S. K., Bray, K. N. C., and Moss, J. B., Combust. Flame 56:199-213 (1984). Bilger, R. W., and Sdrner, S. H., Combust. Flame 51:155-176 (1983). Janicka, J., and Kollmann, W., Seventeenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1979, p. 421. Burke, S. P., and Schumann, T. E. W., Ind. Eng'g Chem. 20:998 (1928). Miller, J. A., Kee, R. J., Smooke, M. D., and Grcar, J. F., Western States Section, The Combustion Institute, Paper WSS/CI 84-10, 1984. Bilger, R. W., Antonia, R. A., and Sreenivasan, K. R., Physics of Fluids 19:1471-1474 (1976). Birch, A. D., Brown, D. R., Dodson, M. G., and Thomas, J. R., J. Fluid Mech. 88:431--449 (1978). Scheffer, R. W., and Dibble, R. W., Sandia Report SAND 85-8837; available from NTIS. Submitted JFM.

285 24.

Antonia, R. A., and Sreenivasan, K. R., Technical Note FM3, 1976, The University of Newcastle, Australia. 25. Lockwood, F. C., and Moneib, H. A., Combust. Sci. Technol. 22:63-81 (1980). 26. Kennedy, I. M., and Kent, J. H., Seventeenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1979, p. 279. 27. Pitz, R. W., and Drake, M. C., 22nd Aerospace Sciences Meeting, Paper AIAA-84-OI97, Jan. 1984. 28. Dibble, R. W., and Hollenbach, R. E., Eighteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1981, p. 1489. 29. Bilger, R. W., in Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), Topics in Applied Physics, Vol. 44, Springer, 1980, p. 65. 30. Corrsin, S., and Kistler, A. L., NACA Report 1244, 1955. 31. Effelsberg, E., and Peters, N., Combust. Flame 50:351-360 (1983). 32. Roberts, P. T., and Moss, J. B., Eighteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1981, p. 941. 33. Williams, F. A., in Turbulent Mixing in Nonreactive and reactive Flows (S. N. B. Murthy, Ed.), p. 189, Plenum, 1975. 34. Peters, N., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1983, p. 353. 35. Rogg, B., Behrendt, F., and Warnatz, J., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh (to appear). 36. Damk6hler, G., Zt. f Elecktroch. 46:601 (1940). [English translation, NACA Tech. Memo, 1112, 1947.]

Received 5 July 1987; revised 10 October 1987