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Cross-correlation of velocity and temperature in a premixed turbulent flame
C O M B U S T I O N A N D F L A M E 51 : 18 3 - 191 ( 19 8 3)
18 3
Cross-Correlation of Velocity and Temperature in a Premixed Turbulent Flame HIDENORI TANAKA and TETSUI YANAGI Department of Applied Physics, The National Defense Academy, Hashirimizu 2-chome 10-20, Yokosuka, 239 Japan
Gradient models of turbulent transport have been widely used without justification in the mathematical modeling of the turbulent flame. The velocity and the temperature of a turbulent premixed flame were measured simultaneously by a laser Doppler anemometer and a compensated fine thermocouple, and their cross-correlation was calculated with a microcomputer. A turbulent heat transport vector was estimated from the measured values in the whole region of the flame. It is directed from the low-temperature side to the high-temperature side in the turbulent reaction zone, which is opposite to the direction expected by the usual gradient transport model and proves the occurrence of the unusual countergradient transport of heat. In the mixing region outside and downstream of the flame it is directed roughly from the high to the low temperature side. However, it was not always transverse to the isotherms but almost parallel in the region near the burner exit, which may suggest that the usual gradient transport assumption is also invalid in the mixing region. Sources of the cross-correlation in both regions were not ascertained from the experimental results, but it was guessed that they may be different between the reaction zone and the outer mixing region.
1. INTRODUCTION
All properties of the turbulent flames, such as velocity, temperature, and composition, fluctuate randomly with time, having complicated interrelations between them. Hence, to obtain a closed set of governing equations for the turbulent combustion phenomena, models for treating interrelation or cross-correlation between fluctuating variables are inevitable. In order to make a reasonable model of cross-correlation between some variables, it is necessary to obtain experimentally their interrelating properties in the flame. Various models for turbulent combustion have been proposed by many researchers. In almost all of these models the conventional gradient or the eddy viscosity model of turbulent transport shown by the equation J~ = - t o , t grad ~ Copyright © 1983 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, NY 10017
(1)
has been used on the analogy of the cold nonreactive flow, where J~ is a turbulent diffusion flux of a scalar quantity ~b, F¢, t an eddy transport coefficient, and ~ the time mean component of ~. The gradient transport model assumes that the turbulent transport of any property occurs in the opposite direction to the gradient of its mean value, which is difficult to justify experimentally in the flames. Libby and Bray [1 ] made a theoretical study of the turbulent premixed flame using their BrayMoss-Libby model, avoiding the conventional gradient transport assumption. They calculated the turbulent transport flux of product for onedimentional flow with turbulent reaction, and they found that it occurred opposite to the direction predicted from the gradient transport equation, that is, the countergradient direction, in the turbulent reaction zone. Many local fluctuating properties of the turbulent flames were measured by using advanced experimental techniques such as a laser Doppler
0010-2180/83/$03.00
184 anemometry, an optical scattering method, and the thermometry with an electrically compensated fine thermocouple. However, only a few measurements of the cross-correlation between some variables were made in the turbulent flames [>51. Moss [3] measured simultaneously the fluctuations of concentration and velocity by a lightscattering technique and the laser Doppler anemometer (LDA), respectively, in the premixed turbulent flame. He estimated the turbulent flux represented by the Favre average from measured values, and he demonstrated the occurrence of the countergradient turbulent flux in the reaction ZO h e .
Yanagi and Mimura [4] estimated the radial velocity-density correlation in a premixed turbulent flame from the measured mean velocity and temperature, using the continuity equation. The estimated correlation had negative value as well as a radial gradient of the mean density in the inner region of the visible flame, which means that a countergradient turbulent transport of mass occurred in the reaction zone. After that they measured the cross-correlation coefficient between fluctuating velocity and temperature in a Bunsen-type flame and proved also the occurrence of the countergradient transport of heat in the turbulent reaction zone [5]. However, the turbulent structure of the flame studied in Ref. [5] was peculiar: The axial velocity component fluctuated rather slowly with large amplitude, but the radial component fluctuated frequently with small amplitude. In this work a turbulent premixed flame with nearly isotropic and rather strong turbulence was investigated to examine the characteristics of the countergradient transport. Simultaneous measurements of the instantaneous velocity and the temperature were made using the LDA and the compensated fine thermocouple in the whole region of the flame. The absolute values of the crosscorrelation between axial or radial velocity and temperature and their coefficients were measured together with mean and fluctuating values. The probability density functions (PDFs) of velocity and temperature were also obtained.
HIDENORI TANAKA and TETSUI YANAGI 2. E X P E R I M E N T A L A P P A R A T U S A N D M E T H O D
The experimental apparatus used in this work was almost the same as that used in Ref. [5], except for a burner shown in Fig. 1. The burner used was constructed carefully so as to produce an axisymmetric flame with nearly isotropic and fairly strong turbulence. Steps within the burner were carefully removed to prevent the deposition of seed particles necessary for the velocity measurement, so that the flow characteristics of the flame were not altered by the addition of seed particles to the flow. The gas mixture passed through a conduit, a contraction with ratio 9 : 1, a turbulence-generating grid, and then a slightly tapered circular tube of 10-ram exit diameter and 50-mm length to produce the nearly isotropic turbulence. A natural gas-air mixture of an equivalence ratio of 0.6 was used as a combustible. The mean velocity was 4.89 m/s (Re 3,200), and the turbulence intensity was about 6% of the mean velocity at the burner exit. The flames was stabilized on the burner by a pilot flame surrounding the burner port, which was made from a slightly rich mixture of natural gas and air. The axial distance x was measured downstream and the radial distance r outward from an origin
I (Naturat gas I .Air)
o
,
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unit:mm
Fig. 1. Burner producing nearly isotropic turbulence.
VELOCITY-TEMPERATURE CORRELATION IN A PREMIXED FLAME at the center of the burner exit. The axial component of an instantaneous velocity is denoted by U = U + u, and the radial component by V = + v, where capital letters with upper bars are the time mean values and the small letters are the fluctuating components. Also, an instantaneous temperature is denoted by T = T + 0. The flow velocity in the flame was measured by the LDA with a tracker-type signal processor. The temperature was measured by a bare PtPt13% Rh thermocouple of 50/~m diam. A thermal lag of the thermocouple was compensated by an electronic circuitry. The upper limit of the compensating frequency was determined to be 2 kHz by a low-pass filter installed into the circuit to reject the high-frequency noise. Each output signal was digitalized simultaneously by a respective A/D converter, and then it was stored in a microcomputer which instantly processed the stored signals. A signal validity circuit was used to interrupt the sampling of both signals when a dropout of the LDA signal occurred. But, except in the extremely outer region of the flame, the dropout was infrequent and the biasing error of the velocity data was negligible. The total number of each signal was about 65,000 and the sampling interval was 0.2 ms. The results obtained from the microcomputer were the PDFs of temperature and velocity, the mean temperature T, the rms value of temperature fluctuation 0', the mean velocity U or V, the rms value of velocity fluctuation u', or u', the cross-correlation uO or v0, and the cross-correlation coefficient R u _ T = uO/u'O' or R v _ w = vO/u'O'. The position of the thermocouple junction put just downstream of the measuring volume of LDA was frequently checked by two telescopes set crosswise. More details of the experimental method are referred to in Ref. [5]. The cross-correlation coefficient was not affected by the mismatch of the compensating time constant [5], but the error in the cross-correlation data was of the same order as the rms value of temperature fluctuation. The method used to determine the time constant of the thermocuple put in the turbulent flame does not seem to be established. In this experiment the time constant
185
of the thermocouple was decided from an oscilloscope trace of the temperature signal after Yoshida [6] as follows. In the middle of the turbulent reaction zone the instantaneous temperature signal showed nearly rectangular fluctuation superposed with small and rapid variation, representative of the wrinkled laminar flame structure of the turbulent premixed flame. The time constant of the compensator was adjusted so that the level of high-temperature interval coincided with the temperature level measured at the highest mean temperature position in the same cross section where the temperature had little fluctuation. A time constant decided by this means was used through the measurements in the same cross section. In the cross sections greater than x/D = 5 (D is the burner exit diameter) the compensating time constant determined in the cross section of x/D = 5 was used, where the center of the reaction zone reached the axis of the flame. The PDFs of temperature at various positions in the cross section of x/D = 2.5 are shown in Fig. 2, where the compenssating time constant was 25 ms. Though the position of the low-temperature (unburned gas) peak moves slightly toward the high-temperature side with increase of r/D, that of the high-temperature (burned gas) peak is almost fixed through the reaction zone where bimodal distribution appears. This fact may suggest that the temperature signal was well compensated during the high-temperature interval, but that it was slightly undercompensated during the low-temperature interval. In the burned gas-air mixing region outside the maximum mean temperature position, the error caused by mismatching the time constant was small because the temperature fluctuated rather slowly. The bimodal shape of the temperature PDF in the reaction zone may mean that the wrinkled laminar flame structure was established in the flame studied. When the temperature of the flame is measured by the thermocouple at the same time as the measurement of velocity by the LDA, it is unavoidable that the fine particles seeded in the stream for the velocity measurement lie on the thermocouple surface. The layer of piled particles obstructs the heat transfer between the thermo-
1 86
HIDENORI TANAKA and TETSUI YANAGI
\ ~ . 0 5
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0 500 1000 1500 T(=K) Fig. 2. Radial distributions of probability density function for instantaneous temperature. x/D = 2.5, compensating time constant ~'e = 25 ms. couple element and the surrounding gas, so that the thermocouple mistakenly represents fluctuating temperature. The change of the various measured values caused by the particle deposition is shown in Fig. 3. The measurement was made at the position in the reaction zone where the cross-correlation between axial velocity and tern-
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perature had a negative maximum value. The left half of the figure shows the change after a new thermocouple was inserted into the flame, and the right half after the piled particles were removed from the wire by a soft brush. It was found that the mean temperature T and the cross-correlation coefficient R u _ r varied slightly, but that the temperature fluctuation 0' and the cross-correlation uO decreased gradually with time. They recovered their initial values when the piled particles were taken off, and after that no significant error arose during a number o f repeated measurements. In this procedure, the thermocouple was cleaned after every three measurements. One measurement took about 1 min. It was also confirmed that the temperature fluctuation measured by this procedure in the seeded flame coincided with that measured previously in the unseeded flame. It may be assumed that output signals o f a constant current hot wire anemometer cannot be compensated by more than ten or fifteen times o f the corner in frequency, due to the nonlinearities involved [7]. The energy-containing frequencies in this experiment were about twenty times of the corner frequency for the thermocouple used. If the thermocouple responded nonlinearly to the large-amplitude fluctuation of temperature, the higher harmonics of the signal would get strong amplification and dominate the high-frequency
VELOCITY-TEMPERATURE CORRELATION IN A PREMIXED FLAME
187
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part of the spectrum. As a supplementary experiment, measurements were made at the center of the reaction zone using a thermocouple of 25 /~m diam which had the time constant of about 7 ms, one third of that of the thermocoupe of 50/~m diam. The cross-correlation and the power spectrum for the thermocouple of 25 /~m diam were not essentially different from those for the
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thermocouple of 50 ~um diam, but the rms value of the temperature fluctuation was slightly increased. Hence, it may be supposed that the compensation of the thermocouple did not cause the serious errors in the high-frequency part of the temperature signal in this experiment. The accuracy of this experiment may be estimated from Fig. 3. The reproducibility of the measured values of temperature and velocity was very satisfactory, but that of the cross-correlation and its coefficient was somewhat inferior: approximately 90% accuracy.
3. EXPERIMENTAL RESULTS 3.1. Structure of the Flame The velocity distribution of the flow without flame at the burner exit (x/D = 0.1) is shown in Fig. 4. The axial component of mean velocity U has a nearly trapezoidal distribution. The turbulence is nearly isotropic and has the intensity of about 6% in the vicinity of the jet axis. The axial component of the velocity fluctuation u' increases with radial distance until a maximum appears above the burner rim, while the radial
HIDENORI TANAKA and TETSUI YANAGI
188
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Fig. 7. Contours of radial mean velocity V (left) and rms values of fluctuating velocity v' (right) of flame.
surrounding the flame axis and decreases sharply outside the flame. The peak of the velocity fluctuation carried by the approach flow appears in the cross sections near the burner exit. It ceases to appear in the region near downstream (x/D = 2), due to the increase of gas viscosity following the temperature rise. The temperature fluctuation becomes violent again in the outer downstream region because of the shear between the burned gas and the ambient air. Figure 7 shows the contours of the radial components of the mean velocity V (left half) and the velocity fluctuation v' (right half). The mean velocity V appears accompanied by the temperature rise and takes its maximum value near the location where the mean temperature is maximum. The radial component of the velocity fluctuation v' has almost the same appearance as the axial component u' except for the lack of the hump of approach flow turbulence. 3.2 Cross-Correlation
component v' decreases. The velocity distribution of the flame at the burner exit (x/D = 0.1) was almost the same as Fig. 4. The structure of the flame studied is shown in Figs. 5-7. Figure 5 shows the contours of the mean temperature T (left half) and the rms value of temperature fluctuation 0' (right half). The outer and inner boundaries of the visible reaction zone determined from a long-exposure photograph are given by dashed lines in this and following figures. A dot-and-dash line in the figure indicates the location where the mean temperature had a maximum value Tmax in the cross section. ~hithin this line both the temperature gradient in the axial direction OT/~x and in the radial direction b7"/Or are positive. In the outer region from the line of Tmax' OT/Oris always negative, while a1"/Ox is either negative in the downstream region or positive in the upstream region. It is seen that the maximum fluctuation of temperature occurred at the middle of the visible reaction zone, where the mean temperature is about 1,000K. Figure 6 shows the contours of the axial components of the mean velocity U (left half) and the velocity fluctuation u' (right half). The mean velocity is almost constant in the wide region
The radial distribution of cross-correlations between temperature and velocity fluctuations in the cross section of x/D = 2.5 is shown together with the mean temperature and rms values of temperature and velocity fluctuations in Fig. 8. In the visible reaction zone the radial velocitytemperature correlation v0 has a fairly large positive peak and the axial velocity-temperature correlation uO has a negative peak, while both correlations have small negative values in the region near the axis. The maxima of both correlations occur at the position where the temperature fluctuation is maximum. They become zero at the location where the mean temperature is maximum and have small positive values in the outer mixing region. The cross-correlation coefficients are also given in the figure. The radial velocity-temperature correlation coefficient Rv-T is similar to that reported in Ref. [5], while the axial one, Ru_T, has a much smaller value than that reported in Ref. [5] in the reaction zone. It is seen from the figure that the velocity fluctuations u' and u' have fairly large values and that no sign of the turbulence generation is seen in the reaction zone,
VELOCITY-TEMPERTURE CORRELATION IN A PREMIXED FLAME
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which differs from that reported in Ref [5]. This difference may be caused by the difference of turbulence in the approach flow. Though the nozzle flow reported in Ref. [5] had weak radial velocity fluctuation and rather strong axial oscillation with low frequency, the approach flow of this experiment had intense and nearly isotropic turbulence with fairly small scale. Accordingly, the generation of turbulence in the reaction zone may be suppressed as predicted by the threeregion model of Ballal [8]. The axial distribution of uO on the flame axis is shown in Fig. 9, which is similar to the radial distribution of vO in the cross section shown in Fig. 8. It shows no correlation at the burner exit, and it takes a negative value at the beginning of the temperature rise (x/D > 2.0). The sign of uO reverses from negative to positive at the center of the visible reaction zone (x/D = 5), where the temperature fluctuation intensity is maximum. The correlation diminshes where the mean temperature becomes maximum, and it grows again in the downstream region. The radial distributions of uO and v0 in the typical cross sections of the flame are given in
189
Fig. 10. The correlation vO has a weak negative value in the region where the mean temperature begins to rise (refer to Fig. 5), and then it has a large positive peak in the visible reaction zone. On the other hand, uO takes a large negative peak in the visible reaction zone of the cross sections near the burner exit. The negative peak moves inward and becomes weaker with increasing distance from the burner exit, and then it ceases to exist in the cross section where the center of the visible reaction zone reaches the flame axis. When the negative peak reaches the flame axis, a new positive peak appears near the outer boundary of the visible reaction zone. It shifts also inward together with the outer edge of the visible reaction zone and ceases at the top o f the contour of the maximum mean temperature Tma x. Both cross-correlations become almost null on the contour Tmax. In the outer mixing region with ambient air they have positive values and become large with the increase of the distance from the burner exit.
4. DISCUSSION The reactant used in this experiment is the lean mixture of air and natural gas of which the major component is methane. The lean mixture of methane and air burns with no change in total mole number. Accordingly, the burned and the unburned gas of the flame studied may produce little change in mean molecular weight by combustion and by mixing with ambient air, and their densities ~ may be approximaded by Oo/T, where Po is a constant. Figure 11 shows the turbulent heat transport vector p vO in the whole region of the flame, where v is a fluctuating component of the velocity vector with components u and v. Dotted lines in the figure indicate the isotherms of the mean temperature and a dot-and-dash line the contour of the maximum mean temperature in the cross section. It may be seen from this figure that the nature of the heat flux inside the contour of maximum mean temperature is different from that on the outside. In the turbulent reaction zone inside the maximum temperature line, except for the region of
190
HIDENORI TANAKA and TETSUI YANAGI
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unburned mixture, the vectors are directed from the low-temperature side to the high-temperature side, which is opposite to the direction expected by the conventional gradient transport model, and this means that the turbulent heat flux occurs
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in the countergradient direction. Moreover, the vectors do not always intersect perpendicularly to the isotherms but are directed individually in complex ways. There is especially violent turbulent heat transfer against the flow in the region near the burner port, where the flow maintains the strong turbulence generated at the burner and the axial mean velocity causes the strong shear. But this intense counterflow turbulent transport may be independent of the shear because its peak moves inward away from the shear region with increasing distance from the burner port. In the outer region of the flame where the burned gas mixes with entrained air, the turbulent heat transport vectors point roughly from the high-temperature side to the low-temperature side. However, as in the reaction zone, they do not always intersect at right angles to the isotherms; but they are almost parallel with isotherms in the upstream region. Hence, though the turbulence in the mixing region is not necessarily isotropic, it may be assumed that the mechanism of turbulent heat transfer governed by the usual gradient-type equation is also invalid in the mixing
VELOCITY-TEMPERATURE
1 -201
CORRELATION
IN A PREMIXED FLAME
(b)
Fig. 12. Comparison of cross-correlation and mean perature gradients. x/D = 2.5: (a) axial distribution (b) radial distribution.
temand
region, which is based on Prandtl’s hypothesis that fluid elements with different temperatures are made to mingle by the small-scale turbulence. The experimental results mentioned above may suggest that the mechanisms of the turbulent heat transfer caused by the cross-correlation of velocity and temperature fluctuations are quite different in the reaction zone and in the mixing region. However, no knowledge of the detailed mechanism of the countergradient diffusion was obtained from this experiment. Libby and Bray [l] explained the origin of the countergradient diffusion of the reaction product as the difference in the partial velocities of reacted and unreacted fluid elements accelerated by the mean pressure difference over the turbulent reaction zone. Moreover, they suggested [9] that the same buoyancy mechanism causes the increase in turbulence in the one dimentional reacting turbulent flow with sufficient heat release. as was the case for the flame studied in this experiment, the heat release parameter of which is about 6. If this mechanism was effective in the flame studied, the turbulence increase would be found on the radial velocity component in the reaction zone of the cross section near the burner exit. However, no sign of the turbulence generation was seen in Figs. 7 and 8, as mentioned before
191
The cross-correlations a and v8 in the cross section x/D = 2.5 are compare.1 \f,rth the corresponding gradients of the mean temperatures aT/ax and a@W, respectively, in Fig. 12. The ratio of cross-correlation to the mean temperature gradient in the axial direction is about 10 times larger than in the radial direction, even though the velocity fluctuations in both directions have the same order of magnitude (see Fig. 4). Moreover, they are different in each cross section. Hence, even if there is some resemblance between their shapes, there is no evidence that the cross-correlation has a direct relation to the mean temperature gradient. The results of this experiment are not compatible with an equation of the form of Eq. (1) in the whole region of the flame studied. Though the mechanism of the turbulent heat transfer in both regions has not been determined, it may become clear from the measurement of the joint PDF between the velocity and the temperature, which is the PDF of the velocity within the group of the fluid elements with the same temperature. The preliminary measurement of the joint PDF suggested that the velocity-temperature correlation in the mixing region is related to the eddy entrainment of the ambient air. The details will be reported in the near future. REFERENCES 1. Libby. P. A., and Bray, K. N. C., AIAA J. 19:205213 (1981). A., and Bray, K. N. C., Sixteenth Sympo2. Ballantyne, sium (International) on Combustion, The Combustion Institute, Pittsburgh, 1977, p. 777. Technot. 22:119-129(1980). 3. Moss,J.B.,Combust.Sci. 4. Yanagi, T., and Mimura, Y., Nensho-Kenkyu 48:1-16 (1978) (in Japanese). Mimura, Y., Yanagi, T., and Matsushita, H., Mem. Nat. Def Acad. Jpn. 20:81-95 (1980). 5. Yanagi, T., and Mimura, Y., Eighteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1981, p. 1031. A., and Tsuji, H., Seventeenth Symposium 6. Yoshida, (International) on Combustion, The Combustion Institute, Pittsburgh, 1979, p. 954. 7. Comte-Bellot, G., Annual Review of Fluid Mechanics, Annual Review Inc., Palo Alto, 1976, p. 209. 8. Ballal,D. R.,Proc. R. Sot. Land. 368:267-282 (1979). 9. Bray, K. N. C., Libby, P. A., Masuya, G., and Moss, J. B., Combust. Sci. Technot. 25:127-140 (1981).
Received 2 7 May 1982; revised 21 December 1982
18 3
Cross-Correlation of Velocity and Temperature in a Premixed Turbulent Flame HIDENORI TANAKA and TETSUI YANAGI Department of Applied Physics, The National Defense Academy, Hashirimizu 2-chome 10-20, Yokosuka, 239 Japan
Gradient models of turbulent transport have been widely used without justification in the mathematical modeling of the turbulent flame. The velocity and the temperature of a turbulent premixed flame were measured simultaneously by a laser Doppler anemometer and a compensated fine thermocouple, and their cross-correlation was calculated with a microcomputer. A turbulent heat transport vector was estimated from the measured values in the whole region of the flame. It is directed from the low-temperature side to the high-temperature side in the turbulent reaction zone, which is opposite to the direction expected by the usual gradient transport model and proves the occurrence of the unusual countergradient transport of heat. In the mixing region outside and downstream of the flame it is directed roughly from the high to the low temperature side. However, it was not always transverse to the isotherms but almost parallel in the region near the burner exit, which may suggest that the usual gradient transport assumption is also invalid in the mixing region. Sources of the cross-correlation in both regions were not ascertained from the experimental results, but it was guessed that they may be different between the reaction zone and the outer mixing region.
1. INTRODUCTION
All properties of the turbulent flames, such as velocity, temperature, and composition, fluctuate randomly with time, having complicated interrelations between them. Hence, to obtain a closed set of governing equations for the turbulent combustion phenomena, models for treating interrelation or cross-correlation between fluctuating variables are inevitable. In order to make a reasonable model of cross-correlation between some variables, it is necessary to obtain experimentally their interrelating properties in the flame. Various models for turbulent combustion have been proposed by many researchers. In almost all of these models the conventional gradient or the eddy viscosity model of turbulent transport shown by the equation J~ = - t o , t grad ~ Copyright © 1983 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, NY 10017
(1)
has been used on the analogy of the cold nonreactive flow, where J~ is a turbulent diffusion flux of a scalar quantity ~b, F¢, t an eddy transport coefficient, and ~ the time mean component of ~. The gradient transport model assumes that the turbulent transport of any property occurs in the opposite direction to the gradient of its mean value, which is difficult to justify experimentally in the flames. Libby and Bray [1 ] made a theoretical study of the turbulent premixed flame using their BrayMoss-Libby model, avoiding the conventional gradient transport assumption. They calculated the turbulent transport flux of product for onedimentional flow with turbulent reaction, and they found that it occurred opposite to the direction predicted from the gradient transport equation, that is, the countergradient direction, in the turbulent reaction zone. Many local fluctuating properties of the turbulent flames were measured by using advanced experimental techniques such as a laser Doppler
0010-2180/83/$03.00
184 anemometry, an optical scattering method, and the thermometry with an electrically compensated fine thermocouple. However, only a few measurements of the cross-correlation between some variables were made in the turbulent flames [>51. Moss [3] measured simultaneously the fluctuations of concentration and velocity by a lightscattering technique and the laser Doppler anemometer (LDA), respectively, in the premixed turbulent flame. He estimated the turbulent flux represented by the Favre average from measured values, and he demonstrated the occurrence of the countergradient turbulent flux in the reaction ZO h e .
Yanagi and Mimura [4] estimated the radial velocity-density correlation in a premixed turbulent flame from the measured mean velocity and temperature, using the continuity equation. The estimated correlation had negative value as well as a radial gradient of the mean density in the inner region of the visible flame, which means that a countergradient turbulent transport of mass occurred in the reaction zone. After that they measured the cross-correlation coefficient between fluctuating velocity and temperature in a Bunsen-type flame and proved also the occurrence of the countergradient transport of heat in the turbulent reaction zone [5]. However, the turbulent structure of the flame studied in Ref. [5] was peculiar: The axial velocity component fluctuated rather slowly with large amplitude, but the radial component fluctuated frequently with small amplitude. In this work a turbulent premixed flame with nearly isotropic and rather strong turbulence was investigated to examine the characteristics of the countergradient transport. Simultaneous measurements of the instantaneous velocity and the temperature were made using the LDA and the compensated fine thermocouple in the whole region of the flame. The absolute values of the crosscorrelation between axial or radial velocity and temperature and their coefficients were measured together with mean and fluctuating values. The probability density functions (PDFs) of velocity and temperature were also obtained.
HIDENORI TANAKA and TETSUI YANAGI 2. E X P E R I M E N T A L A P P A R A T U S A N D M E T H O D
The experimental apparatus used in this work was almost the same as that used in Ref. [5], except for a burner shown in Fig. 1. The burner used was constructed carefully so as to produce an axisymmetric flame with nearly isotropic and fairly strong turbulence. Steps within the burner were carefully removed to prevent the deposition of seed particles necessary for the velocity measurement, so that the flow characteristics of the flame were not altered by the addition of seed particles to the flow. The gas mixture passed through a conduit, a contraction with ratio 9 : 1, a turbulence-generating grid, and then a slightly tapered circular tube of 10-ram exit diameter and 50-mm length to produce the nearly isotropic turbulence. A natural gas-air mixture of an equivalence ratio of 0.6 was used as a combustible. The mean velocity was 4.89 m/s (Re 3,200), and the turbulence intensity was about 6% of the mean velocity at the burner exit. The flames was stabilized on the burner by a pilot flame surrounding the burner port, which was made from a slightly rich mixture of natural gas and air. The axial distance x was measured downstream and the radial distance r outward from an origin
I (Naturat gas I .Air)
o
,
28
Na ~i!a I gas
unit:mm
Fig. 1. Burner producing nearly isotropic turbulence.
VELOCITY-TEMPERATURE CORRELATION IN A PREMIXED FLAME at the center of the burner exit. The axial component of an instantaneous velocity is denoted by U = U + u, and the radial component by V = + v, where capital letters with upper bars are the time mean values and the small letters are the fluctuating components. Also, an instantaneous temperature is denoted by T = T + 0. The flow velocity in the flame was measured by the LDA with a tracker-type signal processor. The temperature was measured by a bare PtPt13% Rh thermocouple of 50/~m diam. A thermal lag of the thermocouple was compensated by an electronic circuitry. The upper limit of the compensating frequency was determined to be 2 kHz by a low-pass filter installed into the circuit to reject the high-frequency noise. Each output signal was digitalized simultaneously by a respective A/D converter, and then it was stored in a microcomputer which instantly processed the stored signals. A signal validity circuit was used to interrupt the sampling of both signals when a dropout of the LDA signal occurred. But, except in the extremely outer region of the flame, the dropout was infrequent and the biasing error of the velocity data was negligible. The total number of each signal was about 65,000 and the sampling interval was 0.2 ms. The results obtained from the microcomputer were the PDFs of temperature and velocity, the mean temperature T, the rms value of temperature fluctuation 0', the mean velocity U or V, the rms value of velocity fluctuation u', or u', the cross-correlation uO or v0, and the cross-correlation coefficient R u _ T = uO/u'O' or R v _ w = vO/u'O'. The position of the thermocouple junction put just downstream of the measuring volume of LDA was frequently checked by two telescopes set crosswise. More details of the experimental method are referred to in Ref. [5]. The cross-correlation coefficient was not affected by the mismatch of the compensating time constant [5], but the error in the cross-correlation data was of the same order as the rms value of temperature fluctuation. The method used to determine the time constant of the thermocuple put in the turbulent flame does not seem to be established. In this experiment the time constant
185
of the thermocouple was decided from an oscilloscope trace of the temperature signal after Yoshida [6] as follows. In the middle of the turbulent reaction zone the instantaneous temperature signal showed nearly rectangular fluctuation superposed with small and rapid variation, representative of the wrinkled laminar flame structure of the turbulent premixed flame. The time constant of the compensator was adjusted so that the level of high-temperature interval coincided with the temperature level measured at the highest mean temperature position in the same cross section where the temperature had little fluctuation. A time constant decided by this means was used through the measurements in the same cross section. In the cross sections greater than x/D = 5 (D is the burner exit diameter) the compensating time constant determined in the cross section of x/D = 5 was used, where the center of the reaction zone reached the axis of the flame. The PDFs of temperature at various positions in the cross section of x/D = 2.5 are shown in Fig. 2, where the compenssating time constant was 25 ms. Though the position of the low-temperature (unburned gas) peak moves slightly toward the high-temperature side with increase of r/D, that of the high-temperature (burned gas) peak is almost fixed through the reaction zone where bimodal distribution appears. This fact may suggest that the temperature signal was well compensated during the high-temperature interval, but that it was slightly undercompensated during the low-temperature interval. In the burned gas-air mixing region outside the maximum mean temperature position, the error caused by mismatching the time constant was small because the temperature fluctuated rather slowly. The bimodal shape of the temperature PDF in the reaction zone may mean that the wrinkled laminar flame structure was established in the flame studied. When the temperature of the flame is measured by the thermocouple at the same time as the measurement of velocity by the LDA, it is unavoidable that the fine particles seeded in the stream for the velocity measurement lie on the thermocouple surface. The layer of piled particles obstructs the heat transfer between the thermo-
1 86
HIDENORI TANAKA and TETSUI YANAGI
\ ~ . 0 5
(]95
0 500 1000 1500 T(=K) Fig. 2. Radial distributions of probability density function for instantaneous temperature. x/D = 2.5, compensating time constant ~'e = 25 ms. couple element and the surrounding gas, so that the thermocouple mistakenly represents fluctuating temperature. The change of the various measured values caused by the particle deposition is shown in Fig. 3. The measurement was made at the position in the reaction zone where the cross-correlation between axial velocity and tern-
,
5
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perature had a negative maximum value. The left half of the figure shows the change after a new thermocouple was inserted into the flame, and the right half after the piled particles were removed from the wire by a soft brush. It was found that the mean temperature T and the cross-correlation coefficient R u _ r varied slightly, but that the temperature fluctuation 0' and the cross-correlation uO decreased gradually with time. They recovered their initial values when the piled particles were taken off, and after that no significant error arose during a number o f repeated measurements. In this procedure, the thermocouple was cleaned after every three measurements. One measurement took about 1 min. It was also confirmed that the temperature fluctuation measured by this procedure in the seeded flame coincided with that measured previously in the unseeded flame. It may be assumed that output signals o f a constant current hot wire anemometer cannot be compensated by more than ten or fifteen times o f the corner in frequency, due to the nonlinearities involved [7]. The energy-containing frequencies in this experiment were about twenty times of the corner frequency for the thermocouple used. If the thermocouple responded nonlinearly to the large-amplitude fluctuation of temperature, the higher harmonics of the signal would get strong amplification and dominate the high-frequency
VELOCITY-TEMPERATURE CORRELATION IN A PREMIXED FLAME
187
-
....
U
U
Q2o~ 2
~
10
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ym~
0.3
, .,,-"6
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x
Fig. 4. Radial distributions of axial and radial components of mean velocity U, V, and rms values of the respective velocity fluctuations u', v', without combustion, x/D = 0.1.
part of the spectrum. As a supplementary experiment, measurements were made at the center of the reaction zone using a thermocouple of 25 /~m diam which had the time constant of about 7 ms, one third of that of the thermocoupe of 50/~m diam. The cross-correlation and the power spectrum for the thermocouple of 25 /~m diam were not essentially different from those for the
T"
xA
0'
unit :'g
NIt/J' Fig. 5. Contours of mean temperature "F (left) and rms values of fluctuating temperature O' (right) of flame: dot-and-dash line, location of maximum mean temperature in the cross section; dotted lines, boundaries of visible reaction zone.
;
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Fig. 6. Contours of axial mean velocity U (left) and rms
valuesof fluctuating velocity u' (right) of flame.
thermocouple of 50 ~um diam, but the rms value of the temperature fluctuation was slightly increased. Hence, it may be supposed that the compensation of the thermocouple did not cause the serious errors in the high-frequency part of the temperature signal in this experiment. The accuracy of this experiment may be estimated from Fig. 3. The reproducibility of the measured values of temperature and velocity was very satisfactory, but that of the cross-correlation and its coefficient was somewhat inferior: approximately 90% accuracy.
3. EXPERIMENTAL RESULTS 3.1. Structure of the Flame The velocity distribution of the flow without flame at the burner exit (x/D = 0.1) is shown in Fig. 4. The axial component of mean velocity U has a nearly trapezoidal distribution. The turbulence is nearly isotropic and has the intensity of about 6% in the vicinity of the jet axis. The axial component of the velocity fluctuation u' increases with radial distance until a maximum appears above the burner rim, while the radial
HIDENORI TANAKA and TETSUI YANAGI
188
, ,-,oL., . 'F-..\°°"
\ \
,0
0
/
0111 o,0
Fig. 7. Contours of radial mean velocity V (left) and rms values of fluctuating velocity v' (right) of flame.
surrounding the flame axis and decreases sharply outside the flame. The peak of the velocity fluctuation carried by the approach flow appears in the cross sections near the burner exit. It ceases to appear in the region near downstream (x/D = 2), due to the increase of gas viscosity following the temperature rise. The temperature fluctuation becomes violent again in the outer downstream region because of the shear between the burned gas and the ambient air. Figure 7 shows the contours of the radial components of the mean velocity V (left half) and the velocity fluctuation v' (right half). The mean velocity V appears accompanied by the temperature rise and takes its maximum value near the location where the mean temperature is maximum. The radial component of the velocity fluctuation v' has almost the same appearance as the axial component u' except for the lack of the hump of approach flow turbulence. 3.2 Cross-Correlation
component v' decreases. The velocity distribution of the flame at the burner exit (x/D = 0.1) was almost the same as Fig. 4. The structure of the flame studied is shown in Figs. 5-7. Figure 5 shows the contours of the mean temperature T (left half) and the rms value of temperature fluctuation 0' (right half). The outer and inner boundaries of the visible reaction zone determined from a long-exposure photograph are given by dashed lines in this and following figures. A dot-and-dash line in the figure indicates the location where the mean temperature had a maximum value Tmax in the cross section. ~hithin this line both the temperature gradient in the axial direction OT/~x and in the radial direction b7"/Or are positive. In the outer region from the line of Tmax' OT/Oris always negative, while a1"/Ox is either negative in the downstream region or positive in the upstream region. It is seen that the maximum fluctuation of temperature occurred at the middle of the visible reaction zone, where the mean temperature is about 1,000K. Figure 6 shows the contours of the axial components of the mean velocity U (left half) and the velocity fluctuation u' (right half). The mean velocity is almost constant in the wide region
The radial distribution of cross-correlations between temperature and velocity fluctuations in the cross section of x/D = 2.5 is shown together with the mean temperature and rms values of temperature and velocity fluctuations in Fig. 8. In the visible reaction zone the radial velocitytemperature correlation v0 has a fairly large positive peak and the axial velocity-temperature correlation uO has a negative peak, while both correlations have small negative values in the region near the axis. The maxima of both correlations occur at the position where the temperature fluctuation is maximum. They become zero at the location where the mean temperature is maximum and have small positive values in the outer mixing region. The cross-correlation coefficients are also given in the figure. The radial velocity-temperature correlation coefficient Rv-T is similar to that reported in Ref. [5], while the axial one, Ru_T, has a much smaller value than that reported in Ref. [5] in the reaction zone. It is seen from the figure that the velocity fluctuations u' and u' have fairly large values and that no sign of the turbulence generation is seen in the reaction zone,
VELOCITY-TEMPERTURE CORRELATION IN A PREMIXED FLAME
o[oo[
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which differs from that reported in Ref [5]. This difference may be caused by the difference of turbulence in the approach flow. Though the nozzle flow reported in Ref. [5] had weak radial velocity fluctuation and rather strong axial oscillation with low frequency, the approach flow of this experiment had intense and nearly isotropic turbulence with fairly small scale. Accordingly, the generation of turbulence in the reaction zone may be suppressed as predicted by the threeregion model of Ballal [8]. The axial distribution of uO on the flame axis is shown in Fig. 9, which is similar to the radial distribution of vO in the cross section shown in Fig. 8. It shows no correlation at the burner exit, and it takes a negative value at the beginning of the temperature rise (x/D > 2.0). The sign of uO reverses from negative to positive at the center of the visible reaction zone (x/D = 5), where the temperature fluctuation intensity is maximum. The correlation diminshes where the mean temperature becomes maximum, and it grows again in the downstream region. The radial distributions of uO and v0 in the typical cross sections of the flame are given in
189
Fig. 10. The correlation vO has a weak negative value in the region where the mean temperature begins to rise (refer to Fig. 5), and then it has a large positive peak in the visible reaction zone. On the other hand, uO takes a large negative peak in the visible reaction zone of the cross sections near the burner exit. The negative peak moves inward and becomes weaker with increasing distance from the burner exit, and then it ceases to exist in the cross section where the center of the visible reaction zone reaches the flame axis. When the negative peak reaches the flame axis, a new positive peak appears near the outer boundary of the visible reaction zone. It shifts also inward together with the outer edge of the visible reaction zone and ceases at the top o f the contour of the maximum mean temperature Tma x. Both cross-correlations become almost null on the contour Tmax. In the outer mixing region with ambient air they have positive values and become large with the increase of the distance from the burner exit.
4. DISCUSSION The reactant used in this experiment is the lean mixture of air and natural gas of which the major component is methane. The lean mixture of methane and air burns with no change in total mole number. Accordingly, the burned and the unburned gas of the flame studied may produce little change in mean molecular weight by combustion and by mixing with ambient air, and their densities ~ may be approximaded by Oo/T, where Po is a constant. Figure 11 shows the turbulent heat transport vector p vO in the whole region of the flame, where v is a fluctuating component of the velocity vector with components u and v. Dotted lines in the figure indicate the isotherms of the mean temperature and a dot-and-dash line the contour of the maximum mean temperature in the cross section. It may be seen from this figure that the nature of the heat flux inside the contour of maximum mean temperature is different from that on the outside. In the turbulent reaction zone inside the maximum temperature line, except for the region of
190
HIDENORI TANAKA and TETSUI YANAGI
_./
.+l
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unburned mixture, the vectors are directed from the low-temperature side to the high-temperature side, which is opposite to the direction expected by the conventional gradient transport model, and this means that the turbulent heat flux occurs
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,%
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in the countergradient direction. Moreover, the vectors do not always intersect perpendicularly to the isotherms but are directed individually in complex ways. There is especially violent turbulent heat transfer against the flow in the region near the burner port, where the flow maintains the strong turbulence generated at the burner and the axial mean velocity causes the strong shear. But this intense counterflow turbulent transport may be independent of the shear because its peak moves inward away from the shear region with increasing distance from the burner port. In the outer region of the flame where the burned gas mixes with entrained air, the turbulent heat transport vectors point roughly from the high-temperature side to the low-temperature side. However, as in the reaction zone, they do not always intersect at right angles to the isotherms; but they are almost parallel with isotherms in the upstream region. Hence, though the turbulence in the mixing region is not necessarily isotropic, it may be assumed that the mechanism of turbulent heat transfer governed by the usual gradient-type equation is also invalid in the mixing
VELOCITY-TEMPERATURE
1 -201
CORRELATION
IN A PREMIXED FLAME
(b)
Fig. 12. Comparison of cross-correlation and mean perature gradients. x/D = 2.5: (a) axial distribution (b) radial distribution.
temand
region, which is based on Prandtl’s hypothesis that fluid elements with different temperatures are made to mingle by the small-scale turbulence. The experimental results mentioned above may suggest that the mechanisms of the turbulent heat transfer caused by the cross-correlation of velocity and temperature fluctuations are quite different in the reaction zone and in the mixing region. However, no knowledge of the detailed mechanism of the countergradient diffusion was obtained from this experiment. Libby and Bray [l] explained the origin of the countergradient diffusion of the reaction product as the difference in the partial velocities of reacted and unreacted fluid elements accelerated by the mean pressure difference over the turbulent reaction zone. Moreover, they suggested [9] that the same buoyancy mechanism causes the increase in turbulence in the one dimentional reacting turbulent flow with sufficient heat release. as was the case for the flame studied in this experiment, the heat release parameter of which is about 6. If this mechanism was effective in the flame studied, the turbulence increase would be found on the radial velocity component in the reaction zone of the cross section near the burner exit. However, no sign of the turbulence generation was seen in Figs. 7 and 8, as mentioned before
191
The cross-correlations a and v8 in the cross section x/D = 2.5 are compare.1 \f,rth the corresponding gradients of the mean temperatures aT/ax and a@W, respectively, in Fig. 12. The ratio of cross-correlation to the mean temperature gradient in the axial direction is about 10 times larger than in the radial direction, even though the velocity fluctuations in both directions have the same order of magnitude (see Fig. 4). Moreover, they are different in each cross section. Hence, even if there is some resemblance between their shapes, there is no evidence that the cross-correlation has a direct relation to the mean temperature gradient. The results of this experiment are not compatible with an equation of the form of Eq. (1) in the whole region of the flame studied. Though the mechanism of the turbulent heat transfer in both regions has not been determined, it may become clear from the measurement of the joint PDF between the velocity and the temperature, which is the PDF of the velocity within the group of the fluid elements with the same temperature. The preliminary measurement of the joint PDF suggested that the velocity-temperature correlation in the mixing region is related to the eddy entrainment of the ambient air. The details will be reported in the near future. REFERENCES 1. Libby. P. A., and Bray, K. N. C., AIAA J. 19:205213 (1981). A., and Bray, K. N. C., Sixteenth Sympo2. Ballantyne, sium (International) on Combustion, The Combustion Institute, Pittsburgh, 1977, p. 777. Technot. 22:119-129(1980). 3. Moss,J.B.,Combust.Sci. 4. Yanagi, T., and Mimura, Y., Nensho-Kenkyu 48:1-16 (1978) (in Japanese). Mimura, Y., Yanagi, T., and Matsushita, H., Mem. Nat. Def Acad. Jpn. 20:81-95 (1980). 5. Yanagi, T., and Mimura, Y., Eighteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1981, p. 1031. A., and Tsuji, H., Seventeenth Symposium 6. Yoshida, (International) on Combustion, The Combustion Institute, Pittsburgh, 1979, p. 954. 7. Comte-Bellot, G., Annual Review of Fluid Mechanics, Annual Review Inc., Palo Alto, 1976, p. 209. 8. Ballal,D. R.,Proc. R. Sot. Land. 368:267-282 (1979). 9. Bray, K. N. C., Libby, P. A., Masuya, G., and Moss, J. B., Combust. Sci. Technot. 25:127-140 (1981).
Received 2 7 May 1982; revised 21 December 1982