Schlieren analysis of an oscillating gas-jet diffusion flame

Schlieren Analysis of an Oscillating Gas-Jet Diffusion Flame BURT W. ALBERS and AJAY K. AGRAWAL*

School of Aerospace and Mechanical Engineering, University of Oklahoma, Norman, OK 73019 USA Flow structure of a flickering gas-jet diffusion flame was investigated using quantitative rainbow schlieren deflectometry. The fuel was pure hydrogen exiting from a burner tube at 43 m/s. Angular deflection data were obtained across the whole field of color schlieren images taken at a spatial resolution of 0.14 mm and a temporal resolution of 60 Hz. These line-of-sight data were analyzed to determine power spectra and flame flicker frequency at different operating pressures to scale the effects of buoyancy. The measurements were tomographically inverted to obtain the refractive index and, hence, temperature distributions assuming chemical equilibrium in the flame. The oscillating flow is described quantitatively in terms of temporal evolution of the temperature field. The flame structure is also described statistically by mean, root-mean-square (RMS), and probability density function profiles of temperature. Results reveal global oscillations in the flow field of the flame. The oscillations were stronger in the outer flow as compared to those at the flame surface. The flame was clipped-off at a downstream location, where the outer vortical structures intensified and entrained cold fluid toward the center region. © 1999 by The Combustion Institute

INTRODUCTION Laminar diffusion flames are known to oscillate or flicker at a low frequency, depending upon the operating conditions. Chamberlin and Rose [1] reported that the flicker frequency of about 10 Hz was not greatly affected by either the fuel type or the burner size tested. Hamins et al. [2] analyzed experimental data over a wide range of operating conditions and found that the flicker frequency correlated with burner size and fuel jet exit velocity. Chen et al. [3] employed planar visualization by reactive Mie-scattering to reveal large toroidal vortices in the oxidizer side of the luminous flame. As these outer vortices convected downstream, they interacted with the flame to create wrinkles on the flame surface. The frequency of the outer vortices correlated with the flame oscillation frequency. The outer vortices develop because of the Kelvin-Helmholtz type instability in the buoyancy-induced shear layer surrounding the flame surface [4]. Detailed numerical simulations [5–11] confirm that the outer vortices develop because of the buoyancy. Furthermore, experiments by Yuan et al. [12] reveal that the flame flicker is suppressed at low buoyancy, achieved by reducing the ambient pressure. Low-gravity experiments by Bahadori et al. [13] show that the flame flicker frequency is proportional to the square root of the gravitational acceleration. *Corresponding author: E-mail: [email protected] 0010-2180/99/$–see front matter PII S0010-2180(99)00034-6

In a flickering flame, the flame surface wrinkles by interactions with the outer vortical flow. This represents a typical feature of turbulent flames, although the flame–flow interactions in the flickering flame are periodic and reproducible. Because of periodic fluctuations, flickering flames serve as a bridge between steady laminar and turbulent flames. For example, analytical models developed from steady flames may be assessed in flickering flames before they are applied to turbulent flames. This characteristic has motivated recent investigations of flickering flames to measure, for example, soot concentration by laser-induced incandescence [14], CO concentration by laser-induced fluorescence imaging [15], and temperature distribution by thinfilament pyrometry [16]. Measurements in these studies were obtained by acoustically forcing the fuel flow rate to phase lock the periodic flame flicker close to the natural flame flicker frequency. Data were acquired simultaneously in 1D using probes or seeding that interfered with the flow. An obstacle to characterizing the unsteady and/or turbulent flows is limitations of experimental diagnostics techniques for spatially and temporally resolved, nonintrusive measurements of temperature and/or species concentrations in 3D flows. Recently, we have applied quantitative rainbow schlieren deflectometry (RSD) [17] to optically measure temperature and/or species concentrations in nonreacting and reacting steady gas flows [18 –24]. In the COMBUSTION AND FLAME 119:84 –94 (1999) © 1999 by The Combustion Institute Published by Elsevier Science Inc.

OSCILLATING GAS-JET DIFFUSION FLAME

Fig. 1. Schematic of the combustion chamber.

present research, we apply the RSD technique to investigate the structure of a flickering gas-jet diffusion flame. Unlike previous studies on flickering flames of hydrocarbons [14 –16] or hydrogen mixed with nitrogen [25, 26], we consider pure hydrogen fuel at a high jet exit velocity. We measure angular deflection across the whole field of color schlieren images. The measured data were reconstructed to obtain the refractive index and, hence, temperature distributions assuming chemical equilibrium in the flame. In the following sections, we describe the experimental setup, provide analytical background, and discuss results. EXPERIMENTAL SYSTEM Experiments at atmospheric and subatmospheric pressures were performed in a continuous-flow combustion chamber of 0.3 m ⫻ 0.3 m cross-section, shown schematically in Fig. 1. The optical access was provided by flat, tempered glass windows (0.2 m ⫻ 0.6 m, and 10 mm thick) on two parallel side walls. The burner was a

85 50-mm-long tube of 1.19 mm diameter and 0.42 mm wall thickness, located at the center of the test section. The effect of side walls on the jet flame is negligible because of the large crosssection of the combustion chamber. Hydrogen was supplied from a compressed gas cylinder, regulated by a needle valve, and measured by a calibrated mass flow meter. Fuel exit velocity at atmospheric pressure was 43.0 ⫾ 1.7 m/s. The corresponding cold jet exit Reynolds number is 500. Room air entered through a pipe and a diffuser filled with glass marbles followed by a honeycomb made of 6-mm-diameter tubes and finally a section of Duocel aluminum foam. Nearly uniform co-air flow with an average velocity of 0.10 ⫾ 0.03 m/s was achieved in the test-section by a vacuum pump located downstream. The co-air flow rate and chamber pressure were controlled independently by two valves near the inlet of the vacuum pump. The co-air flow is considered nearly quiescent because the co-air flow velocity is two orders of magnitude smaller than the fuel jet velocity. At test conditions, the flame was attached to the burner wall in spite of the high fuel jet exit velocity. The flame was visualized using the rainbow schlieren apparatus configured on an optical breadboard mounted on the combustion chamber. Figure 2 shows a schematic of the optical arrangement. The source was a 50-␮m-wide aperture, receiving light via fiber-optic cable from a halogen lamp. The light was collimated and decollimated by achromatic lenses of 63 mm diameter and 490 mm focal length. A magnification lens of 50 mm focal length was used to increase the effective focal length for decollimation to approximately 2700 mm. The schlieren filter was a 35-mm slide with a 12-mmwide symmetric strip of continuously varying colors. The color schlieren images were acquired at a rate of 30 frames per second by a 3-chip CCD camera using a shutter speed of 1/60 second. This system provides data acquisition rate of 60 Hz because each image frame is composed of two image fields taken 1/60 second apart. The acquired image frames were recorded in S-VHS format and later digitized by a 24-bit color frame grabber. The digitized images were stored as 640 ⫻ 480 pixel files in Tag Image File Format (TIFF). Spatial resolution in

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B. W. ALBERS AND A. K. AGRAWAL

Fig. 2. Optical configuration of the rainbow schlieren apparatus. (A) aperture, (B) collimating lens, (C) decollimating lens, (D) magnification lens, (E) rainbow filter, (F) camera lens.

the experiment was 0.14 mm. The field of view was limited to about 40 mm above the burner exit.

Assuming ideal gas law, the refractive index difference is related to mixture properties [20];

␦⫽ ANALYSIS PROCEDURE In a schlieren apparatus, the transverse ray displacement at the filter plane is given as: d ⫽ f c␧共 y兲

(1)

where ƒc is the focal length of the decollimating lens, and ␧(y) is the angular deflection at the projected radial location y. The transverse ray displacement is proportional to the difference between local hue (or color) and background hue in the color schlieren image [17]. The refractive index is found from the Abel inversion of angular deflection data, i.e., ⬁

␦ 共r兲 ⫽ ⫺

1 r

冕冑

␧共 y兲

共 y 2 ⫺ r 2兲

䡠 dy

(2)

r

where ␦ ⫽ (␩ ⫺ 1) is the refractive index difference, and ␩ is the refractive index of the test medium normalized by that of the surrounding air. The integral in Eq. 2 is replaced by a discrete form as

冘D ␧ N

␦ 共r i兲 ⫽ ␦ 共i⌬r兲 ⫽

ij j

(3)

j⫽i

where summation is taken from radial node i to the ambient node N. Using two-point integration, the geometric coefficients Dij are obtained as D ij ⫽ ⫺

1 1 䡠 ␲ 冑共 j ⫹ 0.5兲 2 ⫺ i 2

(4)

P RT

冘 ␬MX l

l

l

(5)

l

Here P is mixture pressure, R is the universal gas constant, and T is mixture temperature. The ␬, M, and X are, respectively, Dale-Gladstone constant, molecular weight, and mole fraction of the species. The summation is taken over all species present in the mixture. In this research, we employ the laminar flamelet concept [27] based on the premise that scalar properties in laminar diffusion flames are nearly universal functions of equivalence ratio or mixture fraction. We assume chemical equilibrium to obtain state relationships and thus, to relate the refractive index difference to the temperature and/or species concentrations using Eq. 5. The validity of this approach in the hydrogen diffusion flame was discussed by Shenoy et al. [21], although the nonunity Lewis number and chemical kinetics effects are important in certain parts of the flame. Chemical equilibrium calculations were performed using Chemkin [28], and Dale-Gladstone constants were taken from Yates [29]. Figure 3 shows temperature as a function of refractive index difference. Different curves in fuel and oxidizer sides of the flame imply that the refractive index is affected significantly by species composition. Refractive index difference is minimum (i.e., temperature is maximum) at the stoichiometric air–fuel ratio. Although not shown here, similar plots relating refractive index difference to species concentrations can be obtained. The quantitative analysis was performed with a set of 100 consecutive color schlieren image frames taken in 3.33 seconds. For analysis, the

OSCILLATING GAS-JET DIFFUSION FLAME

Fig. 3. Temperature versus refractive index difference in hydrogen–air combustion system assuming chemical equilibrium.

color filter calibration curve was approximated by a linear fit with a slope of 53.102 degrees/ mm, i.e., a transverse ray displacement of 1 mm at the filter plane produced a change in hue of 53.102 degrees in the schlieren image. The instantaneous schlieren data were processed to obtain distributions of angular deflection, refractive index difference, and temperature. Angular deflection and refractive index difference were obtained, respectively, from Eqs. 1 and 3. Temperature was found by sixth-order polynomials used to curve fit, respectively, the fuellean and fuel-rich regions in Fig. 3. First, the radial location (rm) of the minimum refractive index (i.e., the maximum temperature), signifying the flame boundary was found at each axial plane. Next, the fuel-rich curve fit was used between the axis and rm and the fuel-lean curve fit was used beyond rm. Only the fuel-lean curve fit was used at axial planes with a maximum temperature far below the adiabatic flame temperature (2400 K). An uncertainty analysis was performed to determine how errors in direct measurements of hue in the color schlieren image affected the measurement accuracy of angular deflection, refractive index difference, and temperature [24]. Using estimated uncertainties of 0.5 degree in local and background hue values and that of 0.5 ␮m in traversing the color filter, the

87 uncertainty in the slope of the filter calibration curve was obtained as 1.498 ⫻ 10⫺3 degree/mm. It corresponded to a maximum uncertainty in angular deflection of 2.85 ⫻ 10-4 degree. An error in angular deflection propagates to uncertainties in refractive index difference, obtained from Eq. 2 using approximation in Eq. 3. For a given error in angular deflection, the inversion error in refractive index difference is the highest at the axis and it decreases with increasing radial distance [20]. In the present experiments, the uncertainty in refractive index difference at the jet axis was 8.0 ⫻ 10-6 and it decreased approximately to 2.0 ⫻ 10-6 near the flame surface. The uncertainty in temperature, obtained using Fig. 3, varies both with the local temperature and stoichiometry. In the oxidizer-side of the flame, the estimated uncertainty in temperature is approximately 150, 100, 50, and 20 K at local temperature of 2400, 2000, 1500, and 1000 K, respectively. The temperature uncertainty in the fuel-side is 2 to 3 times higher than that in the oxidizer-side because of the high sensitivity of temperature to refractive index difference (see Fig. 3) and due to greater inversion errors in the center region. Additionally, the accuracy of Abel inversion in the center region (i.e., the fuel-side of the flame) was compromised by the imperfect symmetry of the jet flow because the jet centerline in an instantaneous schlieren image was only within ⫾0.5 mm of that at the burner exit. Accordingly, the present measurements of temperature in the fuel-side of the flame are considered as qualitative. RESULTS AND DISCUSSION In this section, flicker characteristics are presented and a quantitative description of temperature field is given by instantaneous, mean, RMS, and probability density function (PDF) profiles. Flicker Characteristics Figure 4 shows contour plots of angular deflection obtained from a sequence of color schlieren images to depict the flicker cycle. Figure 4 reveals several important features of the dynamic flame. Angular deflection varies throughout the flame during the flicker cycle, although

88

B. W. ALBERS AND A. K. AGRAWAL

Fig. 5. Time traces of angular deflection at z/d ⫽ 10.

Fig. 4. Contours of angular deflection during the flicker cycle. The plots are 1/60th of a second apart. The contour levels are in 1/100th of a degree.

fluctuations are dominant at z/d ⬎ 15. The contour plot sequence in Figs. 4b to 4e depicts the development of a radically outward bulge at approximately z/d ⫽ 25, suggesting an outer vortex interacting with the flame. Note that previous studies reported flame–vortex interaction occurring typically at z/d ⬍ 10. This observed difference is attributed to the high fuel jet exit velocity (43 m/s) used in the present experiment. In hydrocarbon flames at high fuel jet exit velocity, the outer structures are small and not easily recognizable [30]. The first and last contour plots in Fig. 4 are similar although not identical, implying that the flicker frequency is approximately 12 Hz. Note that angular deflection depicted in Fig. 4 is a path-integrated quantity and, hence, it is indirectly related to the temperature field of the flame by Eqs. 2 and 5. Accordingly, the flame surface cannot be visually identified from Fig. 4.

Further analysis of flame flicker process is presented in Fig. 5 using time traces of angular deflection at z/d ⫽ 10. Evidently, the flow at this axial plane oscillated predominantly between r/d ⫽ 4 and 10. Flow oscillations are not observed in the center region, in part because of small temperature gradients and high fuel jet velocity. The flow did not oscillate at r/d ⱖ 12, which is beyond the heated region of the flame. The time traces in Fig. 5 are highly periodic. The angular deflection data at several axial and radial locations were analyzed using the Fourier transform. The power spectra revealed a dominant frequency that was independent of location in the flow field. Figure 6 shows power spectra at selected locations in the flow field. A frequency of 12.5 ⫾ 0.2 Hz was obtained at all locations with a detectable signal. This result agrees with multipoint laser Doppler velocimetry measurements of Lingens et al. [31, 32], who observed oscillations at a pure frequency in the flow field of flickering flame except for a cone-shaped region near the burner exit. The measured value of flame flicker frequency agrees with that reported in other studies [1, 7, 13, 25, 26, 30 –32] and that obtained from the Strouhal number–Froude number correlation given by Hamins et al. [2]. Experiments were performed at subatmospheric pressures to scale the effects of buoy-

OSCILLATING GAS-JET DIFFUSION FLAME

89

Fig. 6. Power spectra of angular deflection.

ancy on flame flicker. Different buoyancy levels were achieved by varying the jet exit Froude number, which is inversely proportional to the gravitational acceleration (g) or square of the ambient pressure (p2) [22]. The flame flicker frequency was found from Fourier analysis of angular deflection data. Results in Fig. 7 show that the flicker frequency (f) varies linearly with operating pressure, i.e., f ⫽ 12.5p. This result

Fig. 8. Temporal evolution of the temperature field at (a) z/d ⫽ 5, (b) z/d ⫽ 10, (c) z/d ⫽ 15, (d) z/d ⫽ 20, (e) z/d ⫽ 25, and (f) z/d ⫽ 30. Contour labels are 1: 500 K; 2: 1000 K; 3: 1500 K; 4: 2000 K.

concurs with low-gravity experiments of Bahadori et al. [13], who found that ƒ ⬀ g0.5. A linear relationship between f and p is recovered because g ⬀ p2 according to pressure scaling. Note that the line in Fig. 7 was forced to pass through zero, assuming the limiting condition that the flame did not flicker at very low pressures. Temporal Evolution of the Temperature Field

Fig. 7. Flame flicker frequency versus operating pressure, d ⫽ 1.19 mm, Re ⫽ 500.

Figure 8 depicts the dynamic characteristics of the flame at axial planes z/d ⫽ 5, 10, 15, 20, 25 and 30. Although measurements were taken for 3.33 seconds, temperature time traces are shown only for 0.50 seconds using 30 consecutive image fields. The starting time is the same at all axial planes because schlieren data were

90 obtained across the whole field. In Fig. 8, the maximum temperature location is marked by the dotted line. This line represents the flame surface except in Figs. 8e and 8f, where the flame is clipped-off. The flow oscillated at all axial planes as shown in Fig. 8. The flow oscillations are highly periodic and undergo a phase shift in the axial direction because of the convective transport of the outer vortical structures. The average phase velocity was found from the phase shift between z/d ⫽ 5 and 35. The resulting value of 0.6 m/s agrees with the vortex convection velocity of 0.8 m/s measured by Chen et al. [3] and that of 0.8 to 1.6 m/s reported by Lingens et al. [32]. Figure 8 shows large oscillations in the outer flow as compared to wrinkles on the flame surface at a maximum temperature of 2400 K, i.e., the stoichiometric value dictated by chemical equilibrium. The spatial amplitude of oscillations increased in the axial direction as the flame expanded radially. The oscillation amplitude of the flame surface at z/d ⫽ 10, 15, and 20 is, respectively, 0.25 d, 0.37 d and 0.50 d or about half of that at the 500 K contour level. At these axial planes, pockets of high temperature fluid appeared near the flame surface as the flame bulged. Vilimpoc and Goss [25] and Grisch et al. [26] made similar observations in hydrogen– nitrogen jet diffusion flame. Katta et al. [9] proposed that chemical nonequilibrium and nonunity Lewis number effects are responsible for variations in flame temperature during the flicker cycle. The increase and decrease in flame temperature observed in the present study is attributed to these effects even though state relations in Fig. 3 assumed chemical equilibrium. Note that the curves in Fig. 3 are expected to remain qualitatively similar even if chemical kinetics and nonunity Lewis number effects were accounted for. Flow oscillations intensified at z/d ⫽ 25 as the maximum temperature decreased to about 1500 K, suggesting a lack of the flame surface. The oscillation amplitude at the 500 K contour level was about 1.10 d compared with that of 0.70 d at the maximum temperature line. The oscillation amplitude increased and the maximum temperature decreased at z/d ⫽ 30 as the outer vortex swept through the flow field. There is no flame at this axial plane and the oscillation amplitude

B. W. ALBERS AND A. K. AGRAWAL at the 500 K and maximum temperature contour levels is about the same. Note that the flame blow-out evident in Fig. 8 is not indicated by Fig. 4. The decrease in temperature leading to the local flame blow-out is caused by entrainment of cooler fluid toward the center region by the outer vortical structures convecting downstream. Flame clip-off by outer vortical structures was reported in several studies of hydrocarbon flames [14 –16, 30] although not in hydrogen–nitrogen flames [25]. Of particular interest is the downstream development of the flame–vortex interactions. Unfortunately this information was not captured in the present study because of the limited field of view. Statistical Description of the Temperature Field Instantaneous details presented in the previous section are not easily obtained in turbulent flows because of the random 3D fluctuations. In such cases, time statistics are more accurate for comparison between experimental data and modeling predictions [33]. Considering flickering flame as the test case, we present a statistical description of the temperature field by mean, RMS, and PDF profiles. Results are shown in Figs. 9 to 13 for axial planes z/d ⫽ 10, 15, 20, 25, and 30 to represent the field of view. At each location, statistical quantities were computed from 200 temperature data reconstructed from schlieren measurements. The PDF profiles were obtained using 20 equal intervals between the local minimum and maximum temperatures and then normalized by the bin interval. Figure 9 shows mean, RMS, and PDF profiles of temperature at z/d ⫽ 10. The mean temperature profile has a shape typical of a steady diffusion flame. The RMS profile shows a peak in the inner shear layer, a minimum at the mean flame surface, and a peak in the oxidizer-side of the mean flame surface. The peak RMS temperature in the fuel and oxidizer sides of the flame is, respectively, 500 K and 390 K. The mean temperature peaks to 2100 K at r/d ⫽ 5.5, where the RMS temperature is 300 K. Small RMS temperature indicates a relatively stable flame surface at this axial location, as evident in Fig. 8b. These observations agree qualitatively and quantitatively with measurements in flickering propane diffusion flames by Roquemore

OSCILLATING GAS-JET DIFFUSION FLAME

Fig. 9. Temperature profiles at z/d ⫽ 10. (a) mean and RMS, (b) PDF at r/d ⫽ 3, (c) PDF at r/d ⫽ 7.

et al. [30]. The PDF profiles (represented by histograms) reflect different characteristics in the fuel and oxidizer sides of the flame. The temperature PDF in the fuel side (Fig. 9b) is single mode with normal distribution. In contrast, the temperature PDF is bimodal at locations outside of the flame surface, for example, at r/d ⫽ 7 in Fig. 9c. The bimodal PDF shape is associated with oscillations in the outer flow, causing the fluid to alternate between hot and cold modes. Fortuitously, these observations of PDF shapes agree with those of Roquemore et al. [30].

91

Fig. 10. Temperature profiles at z/d ⫽ 15. (a) mean and RMS, (b) PDF at r/d ⫽ 3, (c) PDF at r/d ⫽ 7.

At z/d ⫽ 15, the mean temperature peaks to 2020 K with a RMS temperature of 380 K (see Fig. 10a). Evidently, the maximum mean temperature decreased because temperature fluctuations increased. The RMS temperature peak is unaffected in the fuel side; but it changed from 390 K to 560 K in the oxidizer side. Thus, the outer flow oscillations at this axial plane are more dominant than those at z/d ⫽ 10. The increased dominance of outer flow oscillations arises from radial movement of the vortical structure driving away from the flame surface as the height increases. The PDF temperature

92

Fig. 11. Temperature profiles at z/d ⫽ 20. (a) mean and RMS, (b) PDF at r/d ⫽ 3, (c) PDF at r/d ⫽ 9.

profiles show normal distribution in the fuel side (Fig. 10b) and bimodal distribution with peaks close to the local minimum and maximum temperatures in the oxidizer side (r/d ⫽ 7). Temperature fluctuations increased at z/d ⫽ 20 as shown in Fig. 11a. The maximum mean temperature is 1750 K with an RMS temperature of 470 K. The peak RMS temperature is 620 K in the oxidizer side, where the mean temperature is about 1600 K. A single mode normal PDF distribution is obtained in the fuel side (Fig. 11b). In the oxidizer side (r/d ⫽ 9), the temperature PDF shape is bimodal with peaks

B. W. ALBERS AND A. K. AGRAWAL

Fig. 12. Temperature profiles at z/d ⫽ 25. (a) mean and RMS, (b) PDF at r/d ⫽ 1, (c) PDF at r/d ⫽ 10.

at the local minimum and maximum. This PDF shape is expected in a system with periodic sinusoidal oscillations. At z/d ⫽ 25, mean and RMS temperatures decrease significantly as the flame is extinguished by interaction with the oscillating outer flow (see Fig. 12a). The mean temperature peaks to 1200 K and the peak RMS temperature is 350 K. The temperature PDF profile is single mode in the inner region (Fig. 12b). At r/d ⫽ 10, hot and cold modes are equally probable, as evident by sharp peaks at temperature boundaries in the PDF profiles. The flame remains clipped-off at z/d ⫽ 30 (see Fig. 13a). The mean

OSCILLATING GAS-JET DIFFUSION FLAME

93 oscillations in the flow field of the flame at a frequency varying linearly with the operating pressure. The peak temperature changed as the flame surface stretched/compressed during the flicker cycle. The oscillations were stronger in the outer flow as compared to wrinkles on the flame surface. The spatial amplitude of oscillations increased in the axial direction when the outer vortex intensified approximately at z/d ⫽ 25. Then, the flame was extinguished locally by entrainment of cold fluid in the center region. The mean temperature profiles had a shape typical of that in a steady diffusion flame. The RMS temperature profiles show peaks in the inner shear layer and in the oxidizer-side of the flame surface. The maximum RMS temperature in the flame exceeded 600 K. The temperature PDF profiles show normal distribution in the fuel side of the flame and bimodal distribution in the vortex region. This work was supported in part by the NASA Microgravity Science and Application Division, Grant NAG3-1594. We wish to thank Mr. S. Cherry for help with the experiments.

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Fig. 13. Temperature profiles at z/d ⫽ 30. (a) mean and RMS, (b) PDF at r/d ⫽ 2, (c) PDF at r/d ⫽ 10.

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Received 10 August 1998; revised 17 February 1999; accepted 23 February 1999