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Heat transfer characteristics of a non-premixed turbulent flame formed in a curved rectangular duct
Heat Transfer Characteristics of a Non-Premixed Turbulent Flame Formed in a Curved Rectangular Duct M. TAGAWA,* F. MATSUBARA, and Y. OHTA
Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan Heat transfer characteristics of a non-premixed turbulent flame formed in a curved rectangular duct (180° bend) were investigated experimentally. Key turbulence quantities of velocity and thermal fields such as Reynolds stress components and turbulent heat fluxes were measured using a combined LDV and fine-wire thermocouple technique. These measurements provided direct evidence of the occurrence of the anomalous phenomenon of counter-gradient heat transfer, which can be ascribed to the presence of a strong pressuregradient in the radial direction of the curved duct. The experiment also revealed that the onset region of this “counter-gradient” diffusion was adjacent to the strong “gradient” diffusion region. The quantitative appraisal of the production terms for the turbulent heat flux showed that the pressure gradient promoted gradient diffusion on the inner-wall (low-pressure) side of the curved-duct flame and caused counter-gradient diffusion on the outer-wall (high-pressure) side. The schlieren photography for visualizing the density field showed a totally different behavior of the burned gas parcels between the high- and low-pressure sides of the flame. The essential mechanism causing the counter-gradient diffusion can be explained by the unique motion of the high-temperature (low-density) gas parcel on the high-pressure side of the flame. High-temperature fluid motions tend to be preferentially damped by the pressure gradient imposed on the flow field. The occurrence of the counter-gradient diffusion phenomenon will of course lead to the collapse of the “gradient-diffusion hypothesis,” on which most conventional turbulence models rely. In such a field, the analogy between heat and mass transfer processes, which holds almost always in normal turbulent passive-scalar transport, can disappear. © 2002 by The Combustion Institute
NOMENCLATURE
( )
p R␣
⬘
pressure cross-correlation coefficient between ␣⬘ and ⬘, ⬅ ␣⬘⬘/␣rmsrms r, , z cylindrical coordinate for curved duct (180° bend) flow, see Fig. 1 T, t temperature and fluctuating temperature U, V velocity components in and y directions u, v fluctuating velocity components in and y directions y distance from inner-wall surface of curved duct (⫽ r ⫺ ri; ri: radius of curvature of inner-wall) fluid density Symbols rms
root-mean-square value of fluctuation, ␣⬘2 ⬅
冑
*Corresponding author: E-mail: [email protected] COMBUSTION AND FLAME 129:151–163 (2002) © 2002 by The Combustion Institute Published by Elsevier Science Inc.
共⬃兲 ⬙ ⬍⬎
Reynolds average (conventional timeaverage) fluctuating component from Reynolds average Favre (density-weighted) average, ⬅ ␣/ fluctuating component from Favre average statistical (Reynolds or Favre) average
INTRODUCTION Many features of important transport phenomena—turbulent heat and mass transfer—in high-temperature turbulent reacting flows such as combustion remain unknown, and it is still very difficult to make them sufficiently clear. The thermal efficiency of various combustors for boilers, industrial furnaces and gas turbines must be consistently increased and their combustion stability and controllability improved while further reducing the quantity of discharged environmental pollutants. Because non-premixed turbulent flames are tolerant of a 0010-2180/02/$–see front matter PII 0010-2180(02)00335-8
152 wide range of operating conditions and feasible for high combustion loads, most practical combustors tend to use non-premixed combustion. To develop universal techniques for burning fuels clean and safely, we need to investigate a non-premixed turbulent flame deductively and understand its nature inductively. The key will be to extract important characteristics of the transport phenomena in the non-premixed turbulent combustion field and to identify the essential features one by one. In general, turbulence will enhance heat and mass transfer greatly. Although it is rare for turbulence to reduce their transfer, this does occur. In this case, heat/mass transfer can occur in such a manner that it climbs up the gradient of the mean temperature/mass-concentration distribution. This unusual phenomenon that runs counter to our intuition is termed “counter-gradient diffusion,” and is known to emerge in turbulent flows with large density difference such as temperature-stratified [1] or combusting [2– 8] flows where temperature should be an active scalar. So far, the characteristics of the counter-gradient diffusion in turbulent combustion have been investigated using relatively simple geometries, and its onset mechanism has been discussed based on both experimental and computational results [3, 5, 9 –13]. In turbulent combustion, the pressure gradient working on the flow field is generally recognized to be a primary factor causing countergradient diffusion. Flow fields in combustors are often associated with pressure gradients because of the swirling of the flow and/or curvature of the convergent/divergent shape of the flow passage. Hence, to extract the essential features of counter-gradient diffusion and to investigate its onset mechanism may contribute largely not only to revealing the nature of turbulent flames but also to resolving practical problems in the development and improvement of various combustors, for example, how to control flame-length and flame-position and to enhance or reduce heat transfer to combustor walls. In the present study, we investigate experimentally the counter-gradient diffusion phenomenon in heat transfer, which emerges when a strong pressure-gradient is being imposed on
M. TAGAWA ET AL. the non-premixed turbulent flame, so as to understand the onset mechanism of the phenomenon and the nature of turbulent combustion. The measurement objective is a non-premixed turbulent flame formed in a curved duct (U-shaped duct/180° bend) with a rectangular cross-section, where a strong pressure-gradient in the radial direction of the curvature is formed in the flow field. The behavior of turbulent heat fluxes—the second-order moments of velocity and temperature fluctuations, and key turbulence quantities in enthalpy transport processes as shown in Appendix—is investigated using a simultaneous velocity and temperature measurement technique, in which a laser Doppler velocimeter (LDV) is combined with a digitally compensated fine-wire thermocouple as reported previously [14]. The measurement will give us direct evidence of the occurrence of counter-gradient diffusion in the heat transfer and enable us to identify the onset region. This leads to greater understanding of the influence of counter-gradient diffusion on the statistical characteristics of various turbulence quantities of velocity and temperature fields. The visualization of the density field by the schlieren technique may also reveal the structural difference between a counter-gradient diffusion region and a normal gradient-diffusion one. Synthesizing the results obtained will elucidate the essential features of non-premixed turbulent combustion subjected to a strong pressure-gradient and will clarify the onset mechanism of counter-gradient heat transfer.
EXPERIMENTAL APPARATUS AND PROCEDURE Figure 1 shows the combustor used, whose channel is a U-shaped (180° bent) duct, and the radius of the curvature at the duct center is rc ⫽ 115 mm. One end is connected to the exit of a wind tunnel. The outer- and inner-walls of the duct are made of steel and water-cooled. The side walls are quartz glass plates 3 mm in thickness. The wind-tunnel exit consists of a contraction and an air passage (80 ⫻ 50 mm2) with a slit-shaped fuel exit (5 ⫻ 50 mm2) located
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Fig. 1. Combustion chamber and coordinates.
at the center of the air passage. The fuel exit is movable as depicted in Fig. 1. From the wind tunnel, a uniform air flow with free stream turbulence was supplied to the combustor. At the combustor inlet, the mean m ⫽ 7.7 m/s, and the free stream velocity was U turbulence was 7% in relative intensity, which was generated by a perforated plate (hole size 5 mm, pitch 7 mm) set 100 mm upstream from the combustor inlet. The fuel used was propane, and its flow rate was 3.0 ᐉ/min. The fuel (pro j at the fuel exit was 0.2 m/s, pane) jet velocity U which was 2.6% of the velocity of the air flow (7.7 m/s) entering the combustor. As a result, the flow structure around the fuel exit was a sort of wake. When calculating the Reynolds num md/ (d: width of the ber of the wake by Rej ⫽ U fuel exit), Rej was 2.5 ⫻ 103. In non-combusting flow measurement, no propane was supplied. The cylindrical coordinates (r, , z) are used to describe the flow field (Fig. 1), and the velocity components in the r, , and z directions are denoted by V, U, and W, respectively.1 The test section for the present study is located 60° from the curved-duct inlet and in the center plane (z ⫽ 0 mm). In the following, we introduce another axis y, which is defined as y ⫽ r ⫺ ri (ri: radius of curvature of inner-wall). The turbulent heat flux— here, we use this term to represent the second-order moment of velocity and temperature fluctuations—will dominate the behavior of a turbulent enthalpy 1
In axisymmetric flows such as a swirling jet flame, velocity components in the (r, , z) directions are usually denoted by (V, W, U).
flux (see Appendix). Hence, we can discuss the energy transport processes by turbulence using the turbulent heat flux instead of the turbulent enthalpy flux. In this experiment, the turbulent heat flux was obtained by the simultaneous velocity and temperature measurement technique, in which a home-made LDV with forward-scattering optics was combined with a fine-wire type-R thermocouple (Pt/Pt13%Rh) 40 m in diameter. In the present system, one personal computer was used to control and execute the measurement processes such as the detection and frequency-analysis of Doppler signals, sampling and response-compensation of the thermocouple output, and secondary compilation of the data collected. The use of one computer facilitates the realization of stable and reliable simultaneous measurements [14]. The light source for the LDV was an aircooled Ar-ion laser whose oscillating frequency and maximum output power are 514.5 nm and 100 mW, respectively. The LDV has a frequency-shifting function for reverse-flow measurement. The measurement volume is 0.2 mm in direction and y and 0.8 mm in z. The scattering particles used were alumina particles 1 m in nominal diameter, and were introduced into the flow field from the most upstream region of the wind tunnel using a cyclone seeder. Because the particles were sparsely distributed in the flow, particle seeding caused few problems. The type-R thermocouple was supported with an L-shaped twin-bore ceramic tube, and was introduced into the flow field through a slit provided on the side wall of a quartz-glass plate.
154 The slit was located about 6 cm downstream of the test section. It was confirmed that the presence of the slit had little influence on the measurements. The hot-junction of the thermocouple was positioned just downstream of the LDV measurement volume and was set as close to it as possible. Then, we need to provide a certain distance between the thermocouple and LDV measurement positions so as not to affect the velocity measurement. A distance of 0.5 mm in the direction was thus determined by systematically investigating the influence of distance on turbulent heat-flux measurements. The response lag of the fine-wire thermocouple was compensated using 11 thermocouple measurements sampled continuously for each Doppler-signal detection [14]. Silica-coating for the thermocouple-wire surface, to prevent a catalytic reaction, was not applied, because the combustion was nearly completed at the test section. This helped the thermocouple response not to deteriorate. Turbulence statistics of velocity and temperature fields were calculated from 1024 data— each of which was comprised of one velocity and 11 temperature measurements obtained simultaneously. The data sampling rate was not high and ranged from 10 to 50 data per second mainly because of the sparse particle seeding mentioned above. Reynolds (conventional) or Favre (density-weighted) averaging was applied to the calculation of statistical values. In the following, fluctuations from the Reynolds ( ) and Favre average 共⬃兲 are denoted by ⬘ and
⬙, respectively. Accordingly, streamwise velocity U, radial-direction velocity V and tempera ⫹ u⬘ ⫽ ture T can be decomposed into U ⫽ U ˜ ⫹ u⬙, V ⫽ V ⫹ v⬘ ⫽ V ˜ ⫹ v⬙ and T ⫽ T ⫹ t⬘ ⫽ U ˜ ⫹ t⬙, respectively. T RESULTS AND DISCUSSION Observation of Flame Configuration
Figure 2 shows a long-exposure direct photograph of the non-premixed turbulent flame formed in the curved duct. The exposure time is a half second. The flame was stable and not easily extinguished. The flame is formed along the middle-plane of the duct (rc ⫽ 115 mm), just in the shape of a concentric circle with the
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Fig. 2. Long-exposure photograph of flame (1/2 s).
curved-duct walls. The shape is kept nearly flat in the z-direction. Because the combustion is nearly completed at the test section (the position 60° from the inlet), the flame shows very weak luminescence there. When observing the borders of light and shade in the photograph, we may see a sharper contrast on the outer-wall (high-pressure) than the inner-wall (low-pressure) side. Turbulence Quantities of Velocity Field In this section, the turbulence characteristics of the combustion field shown in Fig. 2 are examined with reference to the isothermal counterpart whose experimental conditions were kept identical to the combustion case. The Reynolds m number based on the uniform flow velocity U and the hydraulic diameter Dh at the combustor mDh/v) was 3.0 ⫻ 104, and the inlet (Re ⬅ U Dean number (De ⬅ Re 公Dh/2rc) 1.6 ⫻ 104. In the present experiment, because the static-pressure distributions on the duct walls were not measured, we evaluated the pressure gradient in the radial direction dp /dr of the isothermal flow 2 using the relation dp /dr ⱌ Um /rc to obtain the 2 value dp /dr ⯝ 6 ⫻ 10 [Pa/m]. Then, the pres2 m sure coefficient Cp ⬅ 2⌬p/U was 1.4. Figures 3, 4, and 5 show the y-direction distributions of the turbulence quantities of the velocity field measured at the duct center (z ⫽ 0 mm). The abscissa shows the distance from the inner wall (y ⫽ 0 mm). Then, the outer wall will be situated at a position y ⫽ 80 mm. shown in The streamwise mean velocity U Fig. 3 has a distribution higher on the inner-wall
FLAME CHARACTERISTICS IN A RECTANGULAR DUCT
and V . Fig. 3. Distributions of mean velocities U
than on the outer-wall side irrespective of the presence of the flame. This indicates the characteristics of an inviscid-fluid flow [15, 16]. The distribution of the combusting flow shows a U gentle peak near the centerline where the flame exists, and is almost uniformly raised to the high-velocity value even in the isothermal regions next to the inner and outer walls. The distribution of the radial-direction velocity V shows that a kind of secondary flow occurs in both of the combusting and non-combusting flows. The flow appears to consist of two flows: one toward the outer wall (y ⬍ 30 mm) and the other toward the inner wall (y ⬎ 40 mm). The relative intensity of these secondary flows is at most 20% of the streamwise mean velocity, and the combustion has little influence on the secondary flows. The isothermal flow shown in Fig. 3 exhibits characteristics generally seen in the curved-duct turbulent flows with square cross-sections [15, 17]. The detailed aspects, however, are naturally different because of differences in the experimental conditions, that is, the aspect ratio of the duct, the presence of the fuel exit, the inflow condition at the duct inlet, and so forth. In general, the behavior of a turbulent flame may be primarily influenced by the characteristics of the fluctuating velocity component normal to the time-averaged flame sheet, that is,
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Fig. 4. Rms velocity distributions urms and vrms.
the fluctuation v⬘ in this experiment. The rootmean square (rms) values of the velocity fluctuations u⬘ and v⬘ in the combusting flow case are shown in Fig. 4 with reference to those of the isothermal flow. As seen from Fig. 4, the rms value of v⬘ (vrms) of the combusting flow shows a marked difference in behavior between the inner- and outer-wall regions that appear to be divided by a border around y ⫽ 50 mm, namely, vrms greatly increases in most of the flameexisting region (15 mm ⬍ y ⬍ 50 mm). In contrast to this, vrms decreases by about 20% around y ⫽ 52 mm in the outer region. The rms value of u⬘ (urms) indicates characteristics similar to vrms, but it behaves in a rather gentle manner. Turbulent combustion is accompanied by large density fluctuations, so that elementary processes such as production, diffusion and dissipation of turbulence energy can usually be discussed on the basis of the Favre-averaged transport equations of turbulence quantities [19 –21]. Because the distribution of the rms ˜ )—v fluctuation from the value of v⬙ (⫽ V ⫺ V Favre average—was very similar to that of vrms (see Fig. 9), we may explain the marked increase of vrms in the combusting flow based on ⬃ the transport equation of v⬙2 . In turbulent com 共⫽ ⫺ bustion fields, v⬙ ⬘v⬘/ 兲 does not become zero. Consequently, when a strong pressure-
156 gradient in the y-direction is imposed on the 共⭸p /⭸y兲 comes to play a flow field, the term ⫺2v⬙ ˜2 . dominant role in the transport equation of v⬙ In the present combustion field, since v⬙ is negative except for the outer narrow region around y ⫽ 50 mm (figure omitted), 共⭸p /⭸y兲 becomes positive almost every⫺2v⬙ where else. Thus, the pressure gradient ⭸p /⭸y results in promoting the production of the turbulence energy of v. In the isothermal flow case, on the other hand, v⬙ is consistently zero, and such a mechanism does not exist. Thus, we may regard the increase of vrms seen in Fig. 4 as “flame-induced turbulence.” We use this term here rather than what is called “flame-generated turbulence,” since the flame-generated turbulence has been investigated in the context of premixed combustion [20, 22, 23]. Further investigation will be needed to confirm whether or not the increase of the rms velocities in the present experiment corresponds to the flamegenerated turbulence. As for the increase in urms, it should be plausible that the increase may be brought about not by the corresponding term 共⭸p /r⭸ 兲 but primarily by the pressure⫺2u⬙ redistribution mechanism between the turbulence energy components, since ⭸p /r⭸ is much smaller than ⭸p /⭸y, and the degree of the increase of urms was weaker than that of vrms. The distributions of the Reynolds-shear stress and the cross-correlation coefficient beu⬘v⬘ tween u⬘ and v⬘, Ruv, in the combustion field are shown in Fig. 5 with reference to the isothermal counterparts. Using the one-dimensional LDV system, we can obtain the Reynolds shear stress from the turbulence statistics measured by rotating the LDV transmission optics on its axis by distri⫾45° [14]. As seen from Fig. 5, the u⬘v⬘ bution of the isothermal flow is consistent with /⭸y (Fig. 3) in terms of the the gradient ⭸U gradient-diffusion hypothesis. However, the consistency is destroyed by the presence of the turns negative in almost the flame, and u⬘v⬘ whole region. This implies that the flow is now dominated by the well-ordered fluid motions comprised of the (u⬘ ⬎ 0, v⬘ ⬍ 0) and (u⬘ ⬍ 0, v⬘ ⬎ 0) flow modules. The fact that the absolute value of Ruv reaches a high value of 0.7 can substantiate the presence of strong regularity in the fluid motions.
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Fig. 5. Reynolds shear stress u⬘v⬘.
Time Constant of Thermocouple When measuring temperature fluctuations with a fine-wire thermocouple, we need to compensate the damped response and the phase lag because of the thermal inertia of the thermocouple wire [24]. Previous studies [14, 25] confirm that the type-R thermocouple 40 m in diameter can reproduce the present turbulent temperature field with sufficient accuracy. The value of the thermocouple time-constant can be estimated from the correlation equation known as the Collis-Williams law [26] for the heat transfer from a fine-wire. Figure 6 shows the result in a contour map as a function of velocity and temperature. To cover the range of velocity
Fig. 6. Time constant of 40 m dia. thermocouple.
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Fig. 8. Correlation coefficients Rut and Rvt.
Fig. 7. Turbulence quantities of thermal field.
and temperature to be measured, we compensated the thermocouple response using the time-constant value 16 ⫾ 4 ms. Turbulence Quantities of Temperature Field Figure 7 shows y-direction distributions of the turbulence quantities of the temperature field, , and v⬘t⬘ together with their possible T , trms, u⬘t⬘ variations because of the change in the thermocouple time-constant (16 ⫾ 4 ms). It is seen from Fig. 7 that the mean temperature distribution T shows a distinct asymmetry. Its gradient is gentle in the inner-wall region and markedly steep on the outer-wall side around y ⫽ 50 mm. Because turbulence mixing will generally contribute to leveling the temperature distribution, the above result implies the presence of active turbulence mixing in the inner-wall region. The distinct increase of vrms in the combustion case (Fig. 4) may also support this. In contrast, turbulence mixing on the outer-wall side is weak or in a rather peculiar state of insulating the high temperature region. It is not necessarily anomalous for the behavior of vrms to be affected by the T distribution, since the temperature in a combustion field is an
active scalar whose behavior cannot be determined unilaterally by a velocity field. The most important feature of the tempera ture field is that the turbulent heat flux v⬘t⬘ (shown in the lower part of Fig. 7) indicates the onset of the counter-gradient diffusion phenomenon. In a normal turbulent temperature can almost field, the qualitative behavior of v⬘t⬘ always be expressed by the gradient-diffusion ⬀ ⫺共⭸T /⭸y兲. As seen from Fig. 7, in model: v⬘t⬘ ⬍ 0 for the inner-wall region of y ⬍ 40 mm, v⬘t⬘ ⭸T /⭸y ⬎ 0, and the gradient-diffusion hypothesis holds there. In the outer-wall region of y ⬎ 46 mm, however, the mean temperature gradient still stays nega⭸T /⭸y turns negative, while v⬘t⬘ tive. This anomaly is hardly observed in normal turbulent heat transfer processes and is called the counter-gradient diffusion. The degree of counter-gradient diffusion observed here is not too strong to transport enthalpy (heat) obviously against the mean-temperature gradient. In other words, little enthalpy (heat) is transported even at the steepest mean-temperature-gradient position around y ⫽ 50 mm, and turbulence itself scarcely contributes to the heat transfer as is the case in laminar heat transfer. The stream stays positive over wise turbulent heat flux u⬘t⬘ the entire region with its peak located at the region, namely, outer edge of the negative v⬘t⬘ the very active turbulence mixing in the innerwall region does not contribute equally to the . production of u⬘t⬘ The cross-correlation coefficients between u⬘ and t⬘, and between v⬘ and t⬘, Rut and Rvt, are shown in Fig. 8. The correlations Rut and Rvt are
158 positive and negative, respectively, in most of the combustion region. From the statistical point of view, there exist strong coherent phenomena in which high-temperature (t⬘ ⬎ 0) fluid parcels tend to move to the inner-wall side (v⬘ ⬍ 0) with high velocity (u⬘ ⬎ 0), and low-temperature (t⬘ ⬍ 0) parcels moving to the outer-wall side (v⬘ ⬎ 0) with low velocity (u⬘ ⬍ 0). It is remarkable that Rvt reaches almost 0.8 widely in the inner-wall region. This shows that v⬘ and t⬘ behave in a highly deterministic way. When recalling Rvt ⯝ ⫺0.5 in normal passive scalar (temperature) transport by turbulence, we need to acknowledge that Rvt ⫽ ⫺0.8 indicates a very strong “gradient-diffusion” heat transfer. Consequently, the strong gradient-diffusion region on the inner-wall side coexists and/or is adjacent to the weak counter-gradient diffusion region on the outer-wall side. The turbulent combustion field in the curved-duct flow can entail these two distinct thermal fields. This is a very unique feature of the turbulent heat-transfer processes in the present combusting flow. When a fuel flow rate was increased to become 1.5 times as large as that of the present experiment, that is 4.5 ᐉ/min, there appeared no essential differences in the heat transfer characteristics from the present results. However, the heat transfer characteristics shown in Figs. 7 and 8 are not always the same. For example, when the fuel exit was moved so as to touch the inner-wall surface, thus keeping the experimental conditions identical to the above case except for the fuel-exit position, we obtained very different results from the present ones. Briefly, the flame was attached to the inner-wall surface and was elongated to the curved-duct exit. The gradient-diffusion region disappeared totally, and “strong” counter-gradient heat transfer emerged at the outer-wall side of the hightemperature region. This suggests that the degree of counter-gradient diffusion can vary depending on the state of the neighboring gradient-diffusion region. Favre-averaged Turbulence Quantities As seen in the conventional turbulent combustion models, Favre averaging can apparently eliminate most of the effects of density fluctua-
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Fig. 9. Favre-averaged turbulence quantities.
tion on the transport equations of turbulence quantities [18, 21]. Among Favre-averaged tur˜2 兲1/ 2 , T ˜2 兲1/ 2 and v⬙t⬙ ˜ ˜ , 共t⬙ bulence quantities, 共v⬙
are of primary importance and are shown in Fig. 9. In Fig. 9, the conventional (Reynolds) averages and the sensitivities to the possible variation in the thermocouple time-constant (16 ⫾ 4 ms, mentioned in the previous section) are included for reference. We estimated the instantaneous fluid density necessary for the Favre-averaging from the instantaneous temperature by treating the fluid as air.2 From Fig. 9, the Favre-averaged values are 2
There is a view that measurement data by the LDV are naturally density-weighted. This is because the seedingparticle number density in a combustion field tends to be in proportion to the gas density even if the initial number density distribution is uniform in the upstream isothermal region. When the measurements are assumed to be densityweighted averages, we may deduce from Fig. 9 that the conventional averages will shift to larger absolute values than those shown in Fig. 9.
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Fig. 10. Schlieren photograph of flame.
smaller than the Reynolds-averaged counterparts by 20 to 30% around y ⫽ 40 mm, where the intensity of the temperature fluctuations is largest. However, we may see little qualitative difference between them, and the aforementioned unique characteristics of the combustion field remain unchanged. Hence, the results obtained may be universal irrespective of which method of statistical averaging is employed. Onset of Counter-Gradient Diffusion and Its Mechanism In this section, we would like to discuss the onset mechanism of the counter-gradient heat transfer based on visual observation of the density field and on a quantitative appraisal of the produc˜ transport equation. tion terms of the v⬙t⬙
Figure 10 shows the instantaneous density field of the combusting flow (Fig. 2) visualized by the schlieren technique. The light source was a xenon flash lamp (Hamamatsu L4643), which provided a light pulse 1 s wide at its half height. The optics were comprised of two-concave mirrors (aperture 150 mm, focal length 1500 mm) and two flat mirrors (aperture 80 mm), and arranged in a so-called Z-shape. The schlieren images were taken by a camera with a telephoto lens of 200 mm focal length, and exposed on black and white film of ISO 400. From Fig. 10, we may see the high-temperature burned gas parcels with large scale turbulent motions move toward the low-pressure
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(inner-wall) side.3 In contrast with this, on the high-pressure (outer-wall) side, a distinct border having a concentric shape with the duct walls was formed between the burned gas and the lowtemperature fluid that can be regarded as air. Furthermore, the motions of the high-temperature gas parcels appear to be rather suppressed, and the low-temperature fluid on the high-pressure side appears to function as a solid wall. In other words, the outer-wall region is in a stably stratified state in which the inner low-density burned gas and the outer high-density air flow are layered in the direction of the centrifugal force as if the low-density gas sat stably on the highdensity one. The fluid motions at such a boundary will be damped. The inner-wall region, on the other hand, is in an unstably stratified state, because the high-density fluid on the inner-wall side sits unstably on the outer low-density burned gas, so that the fluid motions induced by the combustion can grow vigorously. The presence of such phenomena is clearly reflected in the vrms (Fig. 4) and Rvt (Fig. 8) distributions. Among the production terms in the turbulent ˜兲 transport equation [5, 27], the heat-flux 共v⬙t⬙ primary terms can be given by
˜兲 ⯝ Prod. 共v⬙t⬙ ˜ ˜ ˜ ⭸V U ˜ ⭸T ⭸p ⫺ ˜ v⬙t⬙ ⫹ ˜ u⬙t⬙ ⫺ t⬙ ⫺ v⬙ 2 ⭸r ⭸r r ⭸r (i)
(ii)
(iv) (1) As for the present measurement, because the relations ⭸/⭸z ⫽ 0 and ⭸/⭸ ⬍⬍ ⭸/⭸r hold, several terms have been omitted to obtain Eq. 1. Figure 11 shows the results of the quantitative evaluation for terms (i)–(iv) in the right-hand side of Eq. 1. The local pressure-gradient of the term (iv) was estimated using the following relation:
冉
(iii)
冊
1 ⭸ ⭸p ˜2 ⫹ ˜ ⯝ U u⬙ 2 ⫺ 关 r 共 V˜ 2 ⫹ ˜ v⬙ 2兲兴 , ⭸r r r ⭸r (2) 3 Both ends of the fuel exit are in contact with the side walls at the combustor inlet (Fig. 1), and the combustion occurs in the immediate neighborhood of the side walls. As a result, the high-temperature burned gas is partly transported by the secondary flows near the side walls. Such phenomena can reflect in the present schlieren image.
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˜. Fig. 11. Production terms of v⬙t⬙
which can be derived from the momentum equation for V [5, 27]. The contribution of the second term in the right-hand side of Eq. 2 was less than 20% of that of the first term. It is noted that direct evaluation of ⭸p /⭸r from the pressures measured on the inner- and outer-walls will be needed for more precise quantification. From Fig. 11, the terms (ii) and (iii) in Eq. 1 are negligible compared with terms (i) and (iv) ˜ /⭸r or which involve the temperature-gradient ⭸T the pressure gradient ⭸p /⭸r. It is important to note that the term (iv) is dominant, and that the term (i) changes its sign at y ⫽ 45 mm where the mean temperature reaches its maximum (Fig. 9). Because t⬙ ¡ 0 in turbulent flows with negligibly small density fluctuation such as turbulent forced convection, the term (iv) including the pressure gradient will vanish and does not ⬃ contribute at all to the production of v⬙t⬙ (even if the pressure gradient is imposed). As a result, the ⬃ behavior of v⬙t⬙ is to be dominated by the term (i). This may explain why the gradient-diffusion hypothesis will generally work well in the prediction of turbulent transport of a passive scalar. In ⫽ 0, on the other hand, combustion fields where t⬙ the term (iv) becomes negative over the entire region as seen in Fig. 11. This term will then promote gradient diffusion in the inner-wall region and suppress it on the outer-wall side. The present combustion field can be characterized by the fact that the pressure gradient imposed on the flow will play a completely different role in the heat transfer processes between the inner- and outer-wall regions. Namely, the imbalance between the terms (i) and (iv) will induce a strong gradient-diffusion
in the inner-wall region and a weak countergradient diffusion on the outer-wall side of the flame. When viewing the flow field (Fig. 1) from a plane perpendicular to the z-axis, we may find the present combustion field to be similar to the sliced part of a swirling jet flame, which is characterized by ⭸/⭸ ⫽ 0 and ⭸/⭸r ⬎⬎ ⭸/⭸z [5, 8]. As shown in the above, however, the heat transfer processes in the curved-duct combustion field exhibit very unique features which have not been observed in the turbulent swirling jet flames. CONCLUSIONS Combustion itself is a very complicated phenomenon, so that simple experimental or computational configurations are often employed to study it. Apart from such a strategy, we focused the present study on a normally latent phenomenon termed the counter-gradient diffusion, and chose the rather complicated field of the curved-duct turbulent flow. By taking such an approach, we aimed at extracting the essential processes of the heat transfer (enthalpy transport) in non-premixed turbulent combustion and tried to identify their features experimentally. When a strong pressure-gradient was imposed on the non-premixed turbulent flame formed in the curved-duct flow, the countergradient diffusion emerged in the heat transfer. The experiment also revealed that its onset region was adjacent to the “strong” gradientdiffusion region. The essential mechanism causing the counter-gradient diffusion can be explained by the unique motion of the hightemperature (low-density) gas parcel on the high-pressure side. Namely, high-temperature fluid motions tend to be preferentially damped by the pressure gradient imposed on the flow field. The onset of the counter-gradient diffusion phenomenon will directly lead to the collapse of the “gradient-diffusion hypothesis,” on which most of the conventional turbulence models rely. Furthermore, if the effects of a pressuregradient on turbulent transport are distinctly different between heat and mass transfer, the
FLAME CHARACTERISTICS IN A RECTANGULAR DUCT analogy between their transfer processes, which almost always holds in normal turbulent passive-scalar transport, can disappear. A fuel-rich region may then be separated from a hightemperature zone that maintains combustion. Such a phenomenon will cause an increase in the amount of unburned fuel emission. The authors are grateful to T. Shimoji for his assistance in conducting the experiment. This work was partially supported by a Grant-in Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan (Grant No. 12650202).
19.
20. 21.
22. 23. 24. 25. 26. 27.
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Komori, S., and Nagata, K., J. Fluid Mech. 326:205 (1996). Moss, J. B., Combust. Sci. Technol. 22:119 (1980). Shepherd, I. G., Moss, J. B., and Bray, K. N. C., Nineteenth Symposium (International) on Combustion, The Combustion Institute, Haifa, Israel, 1982, p. 423. Tanaka, H., and Yanagi, T., Combust. Flame 51:183 (1983). Takagi, T., Okamoto, T., Taji, M., and Nakasuji, Y., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 251. Heitor, M. V., Taylor, A. M. K. P., and Whitelaw, J. H., J. Fluid Mech. 181:387 (1987). Goss, L. P., Trump, D. D., and Roquemore, W. M., Exp. Fluids 6:189 (1988). Hardalupas, Y., Tagawa, M., and Taylor, A. M. K. P., in Developments in Laser Techniques and Applications to Fluid Mechanics (R. J. Adrian et al., Eds.), SpringerVerlag, Berlin, 1996, p. 159. Libby, P. A., and Bray, K. N. C., AIAA J. 19:205 (1981). Bray, K. N. C., Libby, P. A., Masuya, G., and Moss, J. B., Combust. Sci. Technol. 25:127 (1981). Zhang, S., and Rutland, C. J., Combust. Flame 102:447 (1995). Veynante, D., and Poinsot, T., Center for Turbulence Research: Annual Research Briefs. NASA Ames Research Center & Stanford University, 1995, p. 273. Veynante, D., Trouve, A., Bray, K. N. C., and Mantel, T., J. Fluid Mech. 332:263 (1997). Tagawa, M., Nagaya, S., and Ohta, Y., Exp. Fluids 30:143 (2001). Humphrey, J. A. C., Whitelaw, J. H., and Yee, G., J. Fluid Mech. 103:443 (1981). Kim, W. J., and Patel, V. C., Trans. ASME: J. Fluids Engng. 116:45 (1994). Taylor, A. M. K. P., Whitelaw, J. H., and Yianneskis, M., Trans. ASME: J. Fluids Engng. 104:350 (1982). Bilger, R. W., in Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), Springer-Verlag, Berlin, 1980, p. 65.
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Stårner, S. H., and Bilger, R. W., Twenty-first Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, p. 1569. Driscoll, J. F., and Gulati, A., Combust. Flame 72:131 (1988). Jones, W. P., in Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), Academic Press, London, 1994, p. 309. Shepherd, I. G., and Moss, J. B., AIAA J. 20:566 (1982). Cheng, R. K., Combust. Sci. Technol. 41:109 (1984). Tagawa, M., and Ohta, Y., Combust. Flame 109:549 (1997). Tagawa, M., Shimoji, T., and Ohta, Y., Review Sci. Instrum. 69:3370 (1998). Collis, D. C., and Williams, M. J., J. Fluid Mech. 6:357 (1959). Bradley, D., Gaskell, P. H., Gu, X. J., Lawes, M., and Scott, M. J., Combust. Flame 115:515 (1998). Williams, F. A., Combustion Theory, The Benjamin/ Cummings Publishing Company, California, 1985, p. 1. Miyauchi, T., Tanahashi, M., Sasaki, K., and Ozeki, T., in Transport Phenomena in Combustion (S. H. Chan, Ed.), Taylor & Francis, New York, 1996, p. 1095. Libby, P. A., and Williams, F. A., in Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), SpringerVerlag, Berlin, 1980, p. 1.
Received 11 June 2001; revised 29 October 2001; accepted 28 November 2001
APPENDIX In the analysis of a multicomponent reacting system such as combustion [28], the energy equation can be expressed by enthalpy h.4 Thus, appropriate modeling of the turbulent enthalpy ⬃ flux u⬙k h⬙ is necessary for predicting combusting flows with turbulence models. In the following, we make clear the relationship between the ⬃ turbulent enthalpy flux u⬙k h⬙ and the turbulent ⬃ heat flux u⬙k t⬙ measured. The energy equation in terms of the enthalpy h can be expressed by [30]: 共 h兲, ⫹ 共 u kh兲, k ⫽
冋
冉
共h兲, k ⫹ D 共 j兲 cp
⫺
冊
册
h 共 y 兲, , , cp j j k k
(3)
4 In the direct numerical simulation of premixed turbulent combustion by Miyauchi et al. [29], the energy equation was expressed explicitly by temperature to avoid the iterative computation for obtaining temperature from enthalpy.
162
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where , and ,k denote the derivatives with respect to time and the spatial coordinate x k , respectively. Then, uk is a velocity component in the xk direction, and , h, , and cp are density, enthalpy, heat conductivity and the constantpressure specific heat of a mixture of gases, respectively. The enthalpy, mass fraction and diffusion coefficient of a chemical species j are denoted by hj, yj, and D(j), respectively, where j ⫽ 1 ⬃ n (n: total number of chemical species involved). The summation convention is applied to the indices k and j. It is noted that the time derivative of pressure, (p),, the effect of radiation, and the heat generation because of fluid viscosity are all neglected in Eq. 3. The following relation holds between enthalpy h and temperature T. h ⫽ y jh j ⫽ y j
冋冕
册
T
c pj dT ⫹ h 0j ,
T0
(4)
where T0 is a reference temperature (298.15 K), and cpj and h0j denote the constant-pressure specific heat and the standard heat of formation of the species j. Then, the mean value of the constant-pressure specific heat cpmj is defined by c pmj ⬅
冉 冕 冊冒 T
c pj dT
共T ⫺ T 0兲.
(5)
T0
(6)
Now, by modeling the turbulent enthalpy flux˜ u⬙ h⬙ using the gradient-diffusion hypothk
esis, we obtain
t ˜ ˜ 共h 兲, k, u ⬙kh⬙ ⫽ ⫺ c pm
(7)
where t is the eddy conductivity for heat. From the density-weighted (Favre) decomposition of ˜ ⫹ t⬙ and yj ⫽ ˜yj ⫹ y⬙j, h ˜ can be T and yj, T ⫽ T expressed by ˜ ⫽ c pmj 关y˜j共T ˜ ⫺ T 0兲 ⫹ ⬃ h y ⬙jt⬙ 兴 ⫹ ˜y jh 0j.
˜ 兲, k ⫹ c pmj 共⬃ y ⬙jt⬙ 兲, k其. ⫹ c pmj ˜y j 共T (9) Now, by rewriting the time-averaged value of the convection term uk h using uk ⫽ u ˜k ⫹ u⬙k ˜ ⫹ h⬙, we obtain and h ⫽ h ˜⫹ u kh ⫽ 共u ˜ kh
⬃ u ⬙kh⬙ 兲.
(10)
On the other hand, the substitution of Eq. 6 into uk h yields ⬃ ⬃ ˜ u kh ⫽ c pmj 关共u ˜ k˜y j ⫹ u ⬙ky ⬙j兲 共T ⫺ T 0兲⫹ ˜y j u ⬙kt⬙ ⬃ ⬃ ⫹u ˜ k y ⬙jt⬙ ⫹ u ⬙ky ⬙jt⬙兴⫹ 共u ˜ k˜y j ⬃ ⫹ u ⬙ky ⬙j兲 h 0j.
(11)
Hence, using Eqs. 8, 10, and 11, the turbulent enthalpy flux˜ u⬙k h⬙ can be decomposed into three sorts of turbulence quantities as follows: ˜ ˜ ˜ kh u ⬙kh⬙ ⫽ u kh/ ⫺ u ⬃ ˜ ⫺ T 0兲 ⫹ h 0j兴 u ⬙ky ⬙j ⫽ 关c pmj 共T ⬃ ⬃ ⫹ c pmj ˜y j u ⬙kt⬙ ⫹ c pmj u ⬙ky ⬙jt⬙ .
The substitution of Eq. 5 into Eq. 4 yields h ⫽ y j 关c pmj 共T ⫺ T 0兲 ⫹ h 0j兴.
t ⬃ ˜ ⫺ T 0兲 ⫹ h 0j兴共 ˜y j兲, k u ⬙kh⬙ ⫽ ⫺ 兵关c pmj 共T c pm
(8)
From the insertion of Eq. 8 into the righthand side of Eq. 7, we obtain a turbulence ⬃ model for u⬙k h⬙ :
(12) Applying the general gradient-diffusion models to the terms on the right-hand side of Eq. 12, we have ⬃ u ⬙ky ⬙j ⫽ ⫺D 共 j兲t 共 ˜y j兲, k
(13)
⬃ ˜ 兲, k u ⬙kt⬙ ⫽ ⫺共 t/ c pm兲 共T
(14)
⬃ ⬃ ⬃ ˜ /˜⑀ 兲 u u ⬙ky ⬙jt⬙ ⫽ ⫺c s 共k ⬙ku ⬙ᐉ 共 y ⬙jt⬙ 兲, ᐉ,
(15)
where D(j)t is the eddy diffusivity. Equation 15 is a conventional turbulence model for the triple ˜, and ˜⑀ denote a model product [21], and cs, k ⬃ constant, turbulence energy 共⫽ u⬙k u⬙k /2兲 and the ˜, respectively. dissipation rate of k Now, we can show a close relationship between the application of the gradient-diffusion
FLAME CHARACTERISTICS IN A RECTANGULAR DUCT ⬃ model. (Eq. 7) to u ⬙kh⬙ and the use of Eqs. 13-15. When substituting Eqs. 13-15 into Eq. 12 and comparing the result with Eq. 9, we may find that all the turbulence quantities on the right-hand side of Eq. 12 must obey the gradient-diffusion hypothesis in order for Eq. 7 to be an adequate model for the turbulent enthalpy flux. Because the contri⬃ bution of the triple product to u ⬙kh⬙ should
163
be generally small, the behavior of the turbulent enthalpy flux will be dominated by ⬃ the turbulent heat and mass fluxes u ⬙k t⬙ ˜ and u ⬙ky ⬙j . Thus, it is very important to know whether or not the behaviors of these turbulence quantities can be adequately modeled with the gradient-diffusion hypothesis to justify the conventional turbulence modeling.
Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan Heat transfer characteristics of a non-premixed turbulent flame formed in a curved rectangular duct (180° bend) were investigated experimentally. Key turbulence quantities of velocity and thermal fields such as Reynolds stress components and turbulent heat fluxes were measured using a combined LDV and fine-wire thermocouple technique. These measurements provided direct evidence of the occurrence of the anomalous phenomenon of counter-gradient heat transfer, which can be ascribed to the presence of a strong pressuregradient in the radial direction of the curved duct. The experiment also revealed that the onset region of this “counter-gradient” diffusion was adjacent to the strong “gradient” diffusion region. The quantitative appraisal of the production terms for the turbulent heat flux showed that the pressure gradient promoted gradient diffusion on the inner-wall (low-pressure) side of the curved-duct flame and caused counter-gradient diffusion on the outer-wall (high-pressure) side. The schlieren photography for visualizing the density field showed a totally different behavior of the burned gas parcels between the high- and low-pressure sides of the flame. The essential mechanism causing the counter-gradient diffusion can be explained by the unique motion of the high-temperature (low-density) gas parcel on the high-pressure side of the flame. High-temperature fluid motions tend to be preferentially damped by the pressure gradient imposed on the flow field. The occurrence of the counter-gradient diffusion phenomenon will of course lead to the collapse of the “gradient-diffusion hypothesis,” on which most conventional turbulence models rely. In such a field, the analogy between heat and mass transfer processes, which holds almost always in normal turbulent passive-scalar transport, can disappear. © 2002 by The Combustion Institute
NOMENCLATURE
( )
p R␣
⬘
pressure cross-correlation coefficient between ␣⬘ and ⬘, ⬅ ␣⬘⬘/␣rmsrms r, , z cylindrical coordinate for curved duct (180° bend) flow, see Fig. 1 T, t temperature and fluctuating temperature U, V velocity components in and y directions u, v fluctuating velocity components in and y directions y distance from inner-wall surface of curved duct (⫽ r ⫺ ri; ri: radius of curvature of inner-wall) fluid density Symbols rms
root-mean-square value of fluctuation, ␣⬘2 ⬅
冑
*Corresponding author: E-mail: [email protected] COMBUSTION AND FLAME 129:151–163 (2002) © 2002 by The Combustion Institute Published by Elsevier Science Inc.
共⬃兲 ⬙ ⬍⬎
Reynolds average (conventional timeaverage) fluctuating component from Reynolds average Favre (density-weighted) average, ⬅ ␣/ fluctuating component from Favre average statistical (Reynolds or Favre) average
INTRODUCTION Many features of important transport phenomena—turbulent heat and mass transfer—in high-temperature turbulent reacting flows such as combustion remain unknown, and it is still very difficult to make them sufficiently clear. The thermal efficiency of various combustors for boilers, industrial furnaces and gas turbines must be consistently increased and their combustion stability and controllability improved while further reducing the quantity of discharged environmental pollutants. Because non-premixed turbulent flames are tolerant of a 0010-2180/02/$–see front matter PII 0010-2180(02)00335-8
152 wide range of operating conditions and feasible for high combustion loads, most practical combustors tend to use non-premixed combustion. To develop universal techniques for burning fuels clean and safely, we need to investigate a non-premixed turbulent flame deductively and understand its nature inductively. The key will be to extract important characteristics of the transport phenomena in the non-premixed turbulent combustion field and to identify the essential features one by one. In general, turbulence will enhance heat and mass transfer greatly. Although it is rare for turbulence to reduce their transfer, this does occur. In this case, heat/mass transfer can occur in such a manner that it climbs up the gradient of the mean temperature/mass-concentration distribution. This unusual phenomenon that runs counter to our intuition is termed “counter-gradient diffusion,” and is known to emerge in turbulent flows with large density difference such as temperature-stratified [1] or combusting [2– 8] flows where temperature should be an active scalar. So far, the characteristics of the counter-gradient diffusion in turbulent combustion have been investigated using relatively simple geometries, and its onset mechanism has been discussed based on both experimental and computational results [3, 5, 9 –13]. In turbulent combustion, the pressure gradient working on the flow field is generally recognized to be a primary factor causing countergradient diffusion. Flow fields in combustors are often associated with pressure gradients because of the swirling of the flow and/or curvature of the convergent/divergent shape of the flow passage. Hence, to extract the essential features of counter-gradient diffusion and to investigate its onset mechanism may contribute largely not only to revealing the nature of turbulent flames but also to resolving practical problems in the development and improvement of various combustors, for example, how to control flame-length and flame-position and to enhance or reduce heat transfer to combustor walls. In the present study, we investigate experimentally the counter-gradient diffusion phenomenon in heat transfer, which emerges when a strong pressure-gradient is being imposed on
M. TAGAWA ET AL. the non-premixed turbulent flame, so as to understand the onset mechanism of the phenomenon and the nature of turbulent combustion. The measurement objective is a non-premixed turbulent flame formed in a curved duct (U-shaped duct/180° bend) with a rectangular cross-section, where a strong pressure-gradient in the radial direction of the curvature is formed in the flow field. The behavior of turbulent heat fluxes—the second-order moments of velocity and temperature fluctuations, and key turbulence quantities in enthalpy transport processes as shown in Appendix—is investigated using a simultaneous velocity and temperature measurement technique, in which a laser Doppler velocimeter (LDV) is combined with a digitally compensated fine-wire thermocouple as reported previously [14]. The measurement will give us direct evidence of the occurrence of counter-gradient diffusion in the heat transfer and enable us to identify the onset region. This leads to greater understanding of the influence of counter-gradient diffusion on the statistical characteristics of various turbulence quantities of velocity and temperature fields. The visualization of the density field by the schlieren technique may also reveal the structural difference between a counter-gradient diffusion region and a normal gradient-diffusion one. Synthesizing the results obtained will elucidate the essential features of non-premixed turbulent combustion subjected to a strong pressure-gradient and will clarify the onset mechanism of counter-gradient heat transfer.
EXPERIMENTAL APPARATUS AND PROCEDURE Figure 1 shows the combustor used, whose channel is a U-shaped (180° bent) duct, and the radius of the curvature at the duct center is rc ⫽ 115 mm. One end is connected to the exit of a wind tunnel. The outer- and inner-walls of the duct are made of steel and water-cooled. The side walls are quartz glass plates 3 mm in thickness. The wind-tunnel exit consists of a contraction and an air passage (80 ⫻ 50 mm2) with a slit-shaped fuel exit (5 ⫻ 50 mm2) located
FLAME CHARACTERISTICS IN A RECTANGULAR DUCT
153
Fig. 1. Combustion chamber and coordinates.
at the center of the air passage. The fuel exit is movable as depicted in Fig. 1. From the wind tunnel, a uniform air flow with free stream turbulence was supplied to the combustor. At the combustor inlet, the mean m ⫽ 7.7 m/s, and the free stream velocity was U turbulence was 7% in relative intensity, which was generated by a perforated plate (hole size 5 mm, pitch 7 mm) set 100 mm upstream from the combustor inlet. The fuel used was propane, and its flow rate was 3.0 ᐉ/min. The fuel (pro j at the fuel exit was 0.2 m/s, pane) jet velocity U which was 2.6% of the velocity of the air flow (7.7 m/s) entering the combustor. As a result, the flow structure around the fuel exit was a sort of wake. When calculating the Reynolds num md/ (d: width of the ber of the wake by Rej ⫽ U fuel exit), Rej was 2.5 ⫻ 103. In non-combusting flow measurement, no propane was supplied. The cylindrical coordinates (r, , z) are used to describe the flow field (Fig. 1), and the velocity components in the r, , and z directions are denoted by V, U, and W, respectively.1 The test section for the present study is located 60° from the curved-duct inlet and in the center plane (z ⫽ 0 mm). In the following, we introduce another axis y, which is defined as y ⫽ r ⫺ ri (ri: radius of curvature of inner-wall). The turbulent heat flux— here, we use this term to represent the second-order moment of velocity and temperature fluctuations—will dominate the behavior of a turbulent enthalpy 1
In axisymmetric flows such as a swirling jet flame, velocity components in the (r, , z) directions are usually denoted by (V, W, U).
flux (see Appendix). Hence, we can discuss the energy transport processes by turbulence using the turbulent heat flux instead of the turbulent enthalpy flux. In this experiment, the turbulent heat flux was obtained by the simultaneous velocity and temperature measurement technique, in which a home-made LDV with forward-scattering optics was combined with a fine-wire type-R thermocouple (Pt/Pt13%Rh) 40 m in diameter. In the present system, one personal computer was used to control and execute the measurement processes such as the detection and frequency-analysis of Doppler signals, sampling and response-compensation of the thermocouple output, and secondary compilation of the data collected. The use of one computer facilitates the realization of stable and reliable simultaneous measurements [14]. The light source for the LDV was an aircooled Ar-ion laser whose oscillating frequency and maximum output power are 514.5 nm and 100 mW, respectively. The LDV has a frequency-shifting function for reverse-flow measurement. The measurement volume is 0.2 mm in direction and y and 0.8 mm in z. The scattering particles used were alumina particles 1 m in nominal diameter, and were introduced into the flow field from the most upstream region of the wind tunnel using a cyclone seeder. Because the particles were sparsely distributed in the flow, particle seeding caused few problems. The type-R thermocouple was supported with an L-shaped twin-bore ceramic tube, and was introduced into the flow field through a slit provided on the side wall of a quartz-glass plate.
154 The slit was located about 6 cm downstream of the test section. It was confirmed that the presence of the slit had little influence on the measurements. The hot-junction of the thermocouple was positioned just downstream of the LDV measurement volume and was set as close to it as possible. Then, we need to provide a certain distance between the thermocouple and LDV measurement positions so as not to affect the velocity measurement. A distance of 0.5 mm in the direction was thus determined by systematically investigating the influence of distance on turbulent heat-flux measurements. The response lag of the fine-wire thermocouple was compensated using 11 thermocouple measurements sampled continuously for each Doppler-signal detection [14]. Silica-coating for the thermocouple-wire surface, to prevent a catalytic reaction, was not applied, because the combustion was nearly completed at the test section. This helped the thermocouple response not to deteriorate. Turbulence statistics of velocity and temperature fields were calculated from 1024 data— each of which was comprised of one velocity and 11 temperature measurements obtained simultaneously. The data sampling rate was not high and ranged from 10 to 50 data per second mainly because of the sparse particle seeding mentioned above. Reynolds (conventional) or Favre (density-weighted) averaging was applied to the calculation of statistical values. In the following, fluctuations from the Reynolds ( ) and Favre average 共⬃兲 are denoted by ⬘ and
⬙, respectively. Accordingly, streamwise velocity U, radial-direction velocity V and tempera ⫹ u⬘ ⫽ ture T can be decomposed into U ⫽ U ˜ ⫹ u⬙, V ⫽ V ⫹ v⬘ ⫽ V ˜ ⫹ v⬙ and T ⫽ T ⫹ t⬘ ⫽ U ˜ ⫹ t⬙, respectively. T RESULTS AND DISCUSSION Observation of Flame Configuration
Figure 2 shows a long-exposure direct photograph of the non-premixed turbulent flame formed in the curved duct. The exposure time is a half second. The flame was stable and not easily extinguished. The flame is formed along the middle-plane of the duct (rc ⫽ 115 mm), just in the shape of a concentric circle with the
M. TAGAWA ET AL.
Fig. 2. Long-exposure photograph of flame (1/2 s).
curved-duct walls. The shape is kept nearly flat in the z-direction. Because the combustion is nearly completed at the test section (the position 60° from the inlet), the flame shows very weak luminescence there. When observing the borders of light and shade in the photograph, we may see a sharper contrast on the outer-wall (high-pressure) than the inner-wall (low-pressure) side. Turbulence Quantities of Velocity Field In this section, the turbulence characteristics of the combustion field shown in Fig. 2 are examined with reference to the isothermal counterpart whose experimental conditions were kept identical to the combustion case. The Reynolds m number based on the uniform flow velocity U and the hydraulic diameter Dh at the combustor mDh/v) was 3.0 ⫻ 104, and the inlet (Re ⬅ U Dean number (De ⬅ Re 公Dh/2rc) 1.6 ⫻ 104. In the present experiment, because the static-pressure distributions on the duct walls were not measured, we evaluated the pressure gradient in the radial direction dp /dr of the isothermal flow 2 using the relation dp /dr ⱌ Um /rc to obtain the 2 value dp /dr ⯝ 6 ⫻ 10 [Pa/m]. Then, the pres2 m sure coefficient Cp ⬅ 2⌬p/U was 1.4. Figures 3, 4, and 5 show the y-direction distributions of the turbulence quantities of the velocity field measured at the duct center (z ⫽ 0 mm). The abscissa shows the distance from the inner wall (y ⫽ 0 mm). Then, the outer wall will be situated at a position y ⫽ 80 mm. shown in The streamwise mean velocity U Fig. 3 has a distribution higher on the inner-wall
FLAME CHARACTERISTICS IN A RECTANGULAR DUCT
and V . Fig. 3. Distributions of mean velocities U
than on the outer-wall side irrespective of the presence of the flame. This indicates the characteristics of an inviscid-fluid flow [15, 16]. The distribution of the combusting flow shows a U gentle peak near the centerline where the flame exists, and is almost uniformly raised to the high-velocity value even in the isothermal regions next to the inner and outer walls. The distribution of the radial-direction velocity V shows that a kind of secondary flow occurs in both of the combusting and non-combusting flows. The flow appears to consist of two flows: one toward the outer wall (y ⬍ 30 mm) and the other toward the inner wall (y ⬎ 40 mm). The relative intensity of these secondary flows is at most 20% of the streamwise mean velocity, and the combustion has little influence on the secondary flows. The isothermal flow shown in Fig. 3 exhibits characteristics generally seen in the curved-duct turbulent flows with square cross-sections [15, 17]. The detailed aspects, however, are naturally different because of differences in the experimental conditions, that is, the aspect ratio of the duct, the presence of the fuel exit, the inflow condition at the duct inlet, and so forth. In general, the behavior of a turbulent flame may be primarily influenced by the characteristics of the fluctuating velocity component normal to the time-averaged flame sheet, that is,
155
Fig. 4. Rms velocity distributions urms and vrms.
the fluctuation v⬘ in this experiment. The rootmean square (rms) values of the velocity fluctuations u⬘ and v⬘ in the combusting flow case are shown in Fig. 4 with reference to those of the isothermal flow. As seen from Fig. 4, the rms value of v⬘ (vrms) of the combusting flow shows a marked difference in behavior between the inner- and outer-wall regions that appear to be divided by a border around y ⫽ 50 mm, namely, vrms greatly increases in most of the flameexisting region (15 mm ⬍ y ⬍ 50 mm). In contrast to this, vrms decreases by about 20% around y ⫽ 52 mm in the outer region. The rms value of u⬘ (urms) indicates characteristics similar to vrms, but it behaves in a rather gentle manner. Turbulent combustion is accompanied by large density fluctuations, so that elementary processes such as production, diffusion and dissipation of turbulence energy can usually be discussed on the basis of the Favre-averaged transport equations of turbulence quantities [19 –21]. Because the distribution of the rms ˜ )—v fluctuation from the value of v⬙ (⫽ V ⫺ V Favre average—was very similar to that of vrms (see Fig. 9), we may explain the marked increase of vrms in the combusting flow based on ⬃ the transport equation of v⬙2 . In turbulent com 共⫽ ⫺ bustion fields, v⬙ ⬘v⬘/ 兲 does not become zero. Consequently, when a strong pressure-
156 gradient in the y-direction is imposed on the 共⭸p /⭸y兲 comes to play a flow field, the term ⫺2v⬙ ˜2 . dominant role in the transport equation of v⬙ In the present combustion field, since v⬙ is negative except for the outer narrow region around y ⫽ 50 mm (figure omitted), 共⭸p /⭸y兲 becomes positive almost every⫺2v⬙ where else. Thus, the pressure gradient ⭸p /⭸y results in promoting the production of the turbulence energy of v. In the isothermal flow case, on the other hand, v⬙ is consistently zero, and such a mechanism does not exist. Thus, we may regard the increase of vrms seen in Fig. 4 as “flame-induced turbulence.” We use this term here rather than what is called “flame-generated turbulence,” since the flame-generated turbulence has been investigated in the context of premixed combustion [20, 22, 23]. Further investigation will be needed to confirm whether or not the increase of the rms velocities in the present experiment corresponds to the flamegenerated turbulence. As for the increase in urms, it should be plausible that the increase may be brought about not by the corresponding term 共⭸p /r⭸ 兲 but primarily by the pressure⫺2u⬙ redistribution mechanism between the turbulence energy components, since ⭸p /r⭸ is much smaller than ⭸p /⭸y, and the degree of the increase of urms was weaker than that of vrms. The distributions of the Reynolds-shear stress and the cross-correlation coefficient beu⬘v⬘ tween u⬘ and v⬘, Ruv, in the combustion field are shown in Fig. 5 with reference to the isothermal counterparts. Using the one-dimensional LDV system, we can obtain the Reynolds shear stress from the turbulence statistics measured by rotating the LDV transmission optics on its axis by distri⫾45° [14]. As seen from Fig. 5, the u⬘v⬘ bution of the isothermal flow is consistent with /⭸y (Fig. 3) in terms of the the gradient ⭸U gradient-diffusion hypothesis. However, the consistency is destroyed by the presence of the turns negative in almost the flame, and u⬘v⬘ whole region. This implies that the flow is now dominated by the well-ordered fluid motions comprised of the (u⬘ ⬎ 0, v⬘ ⬍ 0) and (u⬘ ⬍ 0, v⬘ ⬎ 0) flow modules. The fact that the absolute value of Ruv reaches a high value of 0.7 can substantiate the presence of strong regularity in the fluid motions.
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Fig. 5. Reynolds shear stress u⬘v⬘.
Time Constant of Thermocouple When measuring temperature fluctuations with a fine-wire thermocouple, we need to compensate the damped response and the phase lag because of the thermal inertia of the thermocouple wire [24]. Previous studies [14, 25] confirm that the type-R thermocouple 40 m in diameter can reproduce the present turbulent temperature field with sufficient accuracy. The value of the thermocouple time-constant can be estimated from the correlation equation known as the Collis-Williams law [26] for the heat transfer from a fine-wire. Figure 6 shows the result in a contour map as a function of velocity and temperature. To cover the range of velocity
Fig. 6. Time constant of 40 m dia. thermocouple.
FLAME CHARACTERISTICS IN A RECTANGULAR DUCT
157
Fig. 8. Correlation coefficients Rut and Rvt.
Fig. 7. Turbulence quantities of thermal field.
and temperature to be measured, we compensated the thermocouple response using the time-constant value 16 ⫾ 4 ms. Turbulence Quantities of Temperature Field Figure 7 shows y-direction distributions of the turbulence quantities of the temperature field, , and v⬘t⬘ together with their possible T , trms, u⬘t⬘ variations because of the change in the thermocouple time-constant (16 ⫾ 4 ms). It is seen from Fig. 7 that the mean temperature distribution T shows a distinct asymmetry. Its gradient is gentle in the inner-wall region and markedly steep on the outer-wall side around y ⫽ 50 mm. Because turbulence mixing will generally contribute to leveling the temperature distribution, the above result implies the presence of active turbulence mixing in the inner-wall region. The distinct increase of vrms in the combustion case (Fig. 4) may also support this. In contrast, turbulence mixing on the outer-wall side is weak or in a rather peculiar state of insulating the high temperature region. It is not necessarily anomalous for the behavior of vrms to be affected by the T distribution, since the temperature in a combustion field is an
active scalar whose behavior cannot be determined unilaterally by a velocity field. The most important feature of the tempera ture field is that the turbulent heat flux v⬘t⬘ (shown in the lower part of Fig. 7) indicates the onset of the counter-gradient diffusion phenomenon. In a normal turbulent temperature can almost field, the qualitative behavior of v⬘t⬘ always be expressed by the gradient-diffusion ⬀ ⫺共⭸T /⭸y兲. As seen from Fig. 7, in model: v⬘t⬘ ⬍ 0 for the inner-wall region of y ⬍ 40 mm, v⬘t⬘ ⭸T /⭸y ⬎ 0, and the gradient-diffusion hypothesis holds there. In the outer-wall region of y ⬎ 46 mm, however, the mean temperature gradient still stays nega⭸T /⭸y turns negative, while v⬘t⬘ tive. This anomaly is hardly observed in normal turbulent heat transfer processes and is called the counter-gradient diffusion. The degree of counter-gradient diffusion observed here is not too strong to transport enthalpy (heat) obviously against the mean-temperature gradient. In other words, little enthalpy (heat) is transported even at the steepest mean-temperature-gradient position around y ⫽ 50 mm, and turbulence itself scarcely contributes to the heat transfer as is the case in laminar heat transfer. The stream stays positive over wise turbulent heat flux u⬘t⬘ the entire region with its peak located at the region, namely, outer edge of the negative v⬘t⬘ the very active turbulence mixing in the innerwall region does not contribute equally to the . production of u⬘t⬘ The cross-correlation coefficients between u⬘ and t⬘, and between v⬘ and t⬘, Rut and Rvt, are shown in Fig. 8. The correlations Rut and Rvt are
158 positive and negative, respectively, in most of the combustion region. From the statistical point of view, there exist strong coherent phenomena in which high-temperature (t⬘ ⬎ 0) fluid parcels tend to move to the inner-wall side (v⬘ ⬍ 0) with high velocity (u⬘ ⬎ 0), and low-temperature (t⬘ ⬍ 0) parcels moving to the outer-wall side (v⬘ ⬎ 0) with low velocity (u⬘ ⬍ 0). It is remarkable that Rvt reaches almost 0.8 widely in the inner-wall region. This shows that v⬘ and t⬘ behave in a highly deterministic way. When recalling Rvt ⯝ ⫺0.5 in normal passive scalar (temperature) transport by turbulence, we need to acknowledge that Rvt ⫽ ⫺0.8 indicates a very strong “gradient-diffusion” heat transfer. Consequently, the strong gradient-diffusion region on the inner-wall side coexists and/or is adjacent to the weak counter-gradient diffusion region on the outer-wall side. The turbulent combustion field in the curved-duct flow can entail these two distinct thermal fields. This is a very unique feature of the turbulent heat-transfer processes in the present combusting flow. When a fuel flow rate was increased to become 1.5 times as large as that of the present experiment, that is 4.5 ᐉ/min, there appeared no essential differences in the heat transfer characteristics from the present results. However, the heat transfer characteristics shown in Figs. 7 and 8 are not always the same. For example, when the fuel exit was moved so as to touch the inner-wall surface, thus keeping the experimental conditions identical to the above case except for the fuel-exit position, we obtained very different results from the present ones. Briefly, the flame was attached to the inner-wall surface and was elongated to the curved-duct exit. The gradient-diffusion region disappeared totally, and “strong” counter-gradient heat transfer emerged at the outer-wall side of the hightemperature region. This suggests that the degree of counter-gradient diffusion can vary depending on the state of the neighboring gradient-diffusion region. Favre-averaged Turbulence Quantities As seen in the conventional turbulent combustion models, Favre averaging can apparently eliminate most of the effects of density fluctua-
M. TAGAWA ET AL.
Fig. 9. Favre-averaged turbulence quantities.
tion on the transport equations of turbulence quantities [18, 21]. Among Favre-averaged tur˜2 兲1/ 2 , T ˜2 兲1/ 2 and v⬙t⬙ ˜ ˜ , 共t⬙ bulence quantities, 共v⬙
are of primary importance and are shown in Fig. 9. In Fig. 9, the conventional (Reynolds) averages and the sensitivities to the possible variation in the thermocouple time-constant (16 ⫾ 4 ms, mentioned in the previous section) are included for reference. We estimated the instantaneous fluid density necessary for the Favre-averaging from the instantaneous temperature by treating the fluid as air.2 From Fig. 9, the Favre-averaged values are 2
There is a view that measurement data by the LDV are naturally density-weighted. This is because the seedingparticle number density in a combustion field tends to be in proportion to the gas density even if the initial number density distribution is uniform in the upstream isothermal region. When the measurements are assumed to be densityweighted averages, we may deduce from Fig. 9 that the conventional averages will shift to larger absolute values than those shown in Fig. 9.
FLAME CHARACTERISTICS IN A RECTANGULAR DUCT
Fig. 10. Schlieren photograph of flame.
smaller than the Reynolds-averaged counterparts by 20 to 30% around y ⫽ 40 mm, where the intensity of the temperature fluctuations is largest. However, we may see little qualitative difference between them, and the aforementioned unique characteristics of the combustion field remain unchanged. Hence, the results obtained may be universal irrespective of which method of statistical averaging is employed. Onset of Counter-Gradient Diffusion and Its Mechanism In this section, we would like to discuss the onset mechanism of the counter-gradient heat transfer based on visual observation of the density field and on a quantitative appraisal of the produc˜ transport equation. tion terms of the v⬙t⬙
Figure 10 shows the instantaneous density field of the combusting flow (Fig. 2) visualized by the schlieren technique. The light source was a xenon flash lamp (Hamamatsu L4643), which provided a light pulse 1 s wide at its half height. The optics were comprised of two-concave mirrors (aperture 150 mm, focal length 1500 mm) and two flat mirrors (aperture 80 mm), and arranged in a so-called Z-shape. The schlieren images were taken by a camera with a telephoto lens of 200 mm focal length, and exposed on black and white film of ISO 400. From Fig. 10, we may see the high-temperature burned gas parcels with large scale turbulent motions move toward the low-pressure
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(inner-wall) side.3 In contrast with this, on the high-pressure (outer-wall) side, a distinct border having a concentric shape with the duct walls was formed between the burned gas and the lowtemperature fluid that can be regarded as air. Furthermore, the motions of the high-temperature gas parcels appear to be rather suppressed, and the low-temperature fluid on the high-pressure side appears to function as a solid wall. In other words, the outer-wall region is in a stably stratified state in which the inner low-density burned gas and the outer high-density air flow are layered in the direction of the centrifugal force as if the low-density gas sat stably on the highdensity one. The fluid motions at such a boundary will be damped. The inner-wall region, on the other hand, is in an unstably stratified state, because the high-density fluid on the inner-wall side sits unstably on the outer low-density burned gas, so that the fluid motions induced by the combustion can grow vigorously. The presence of such phenomena is clearly reflected in the vrms (Fig. 4) and Rvt (Fig. 8) distributions. Among the production terms in the turbulent ˜兲 transport equation [5, 27], the heat-flux 共v⬙t⬙ primary terms can be given by
˜兲 ⯝ Prod. 共v⬙t⬙ ˜ ˜ ˜ ⭸V U ˜ ⭸T ⭸p ⫺ ˜ v⬙t⬙ ⫹ ˜ u⬙t⬙ ⫺ t⬙ ⫺ v⬙ 2 ⭸r ⭸r r ⭸r (i)
(ii)
(iv) (1) As for the present measurement, because the relations ⭸/⭸z ⫽ 0 and ⭸/⭸ ⬍⬍ ⭸/⭸r hold, several terms have been omitted to obtain Eq. 1. Figure 11 shows the results of the quantitative evaluation for terms (i)–(iv) in the right-hand side of Eq. 1. The local pressure-gradient of the term (iv) was estimated using the following relation:
冉
(iii)
冊
1 ⭸ ⭸p ˜2 ⫹ ˜ ⯝ U u⬙ 2 ⫺ 关 r 共 V˜ 2 ⫹ ˜ v⬙ 2兲兴 , ⭸r r r ⭸r (2) 3 Both ends of the fuel exit are in contact with the side walls at the combustor inlet (Fig. 1), and the combustion occurs in the immediate neighborhood of the side walls. As a result, the high-temperature burned gas is partly transported by the secondary flows near the side walls. Such phenomena can reflect in the present schlieren image.
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˜. Fig. 11. Production terms of v⬙t⬙
which can be derived from the momentum equation for V [5, 27]. The contribution of the second term in the right-hand side of Eq. 2 was less than 20% of that of the first term. It is noted that direct evaluation of ⭸p /⭸r from the pressures measured on the inner- and outer-walls will be needed for more precise quantification. From Fig. 11, the terms (ii) and (iii) in Eq. 1 are negligible compared with terms (i) and (iv) ˜ /⭸r or which involve the temperature-gradient ⭸T the pressure gradient ⭸p /⭸r. It is important to note that the term (iv) is dominant, and that the term (i) changes its sign at y ⫽ 45 mm where the mean temperature reaches its maximum (Fig. 9). Because t⬙ ¡ 0 in turbulent flows with negligibly small density fluctuation such as turbulent forced convection, the term (iv) including the pressure gradient will vanish and does not ⬃ contribute at all to the production of v⬙t⬙ (even if the pressure gradient is imposed). As a result, the ⬃ behavior of v⬙t⬙ is to be dominated by the term (i). This may explain why the gradient-diffusion hypothesis will generally work well in the prediction of turbulent transport of a passive scalar. In ⫽ 0, on the other hand, combustion fields where t⬙ the term (iv) becomes negative over the entire region as seen in Fig. 11. This term will then promote gradient diffusion in the inner-wall region and suppress it on the outer-wall side. The present combustion field can be characterized by the fact that the pressure gradient imposed on the flow will play a completely different role in the heat transfer processes between the inner- and outer-wall regions. Namely, the imbalance between the terms (i) and (iv) will induce a strong gradient-diffusion
in the inner-wall region and a weak countergradient diffusion on the outer-wall side of the flame. When viewing the flow field (Fig. 1) from a plane perpendicular to the z-axis, we may find the present combustion field to be similar to the sliced part of a swirling jet flame, which is characterized by ⭸/⭸ ⫽ 0 and ⭸/⭸r ⬎⬎ ⭸/⭸z [5, 8]. As shown in the above, however, the heat transfer processes in the curved-duct combustion field exhibit very unique features which have not been observed in the turbulent swirling jet flames. CONCLUSIONS Combustion itself is a very complicated phenomenon, so that simple experimental or computational configurations are often employed to study it. Apart from such a strategy, we focused the present study on a normally latent phenomenon termed the counter-gradient diffusion, and chose the rather complicated field of the curved-duct turbulent flow. By taking such an approach, we aimed at extracting the essential processes of the heat transfer (enthalpy transport) in non-premixed turbulent combustion and tried to identify their features experimentally. When a strong pressure-gradient was imposed on the non-premixed turbulent flame formed in the curved-duct flow, the countergradient diffusion emerged in the heat transfer. The experiment also revealed that its onset region was adjacent to the “strong” gradientdiffusion region. The essential mechanism causing the counter-gradient diffusion can be explained by the unique motion of the hightemperature (low-density) gas parcel on the high-pressure side. Namely, high-temperature fluid motions tend to be preferentially damped by the pressure gradient imposed on the flow field. The onset of the counter-gradient diffusion phenomenon will directly lead to the collapse of the “gradient-diffusion hypothesis,” on which most of the conventional turbulence models rely. Furthermore, if the effects of a pressuregradient on turbulent transport are distinctly different between heat and mass transfer, the
FLAME CHARACTERISTICS IN A RECTANGULAR DUCT analogy between their transfer processes, which almost always holds in normal turbulent passive-scalar transport, can disappear. A fuel-rich region may then be separated from a hightemperature zone that maintains combustion. Such a phenomenon will cause an increase in the amount of unburned fuel emission. The authors are grateful to T. Shimoji for his assistance in conducting the experiment. This work was partially supported by a Grant-in Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan (Grant No. 12650202).
19.
20. 21.
22. 23. 24. 25. 26. 27.
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Komori, S., and Nagata, K., J. Fluid Mech. 326:205 (1996). Moss, J. B., Combust. Sci. Technol. 22:119 (1980). Shepherd, I. G., Moss, J. B., and Bray, K. N. C., Nineteenth Symposium (International) on Combustion, The Combustion Institute, Haifa, Israel, 1982, p. 423. Tanaka, H., and Yanagi, T., Combust. Flame 51:183 (1983). Takagi, T., Okamoto, T., Taji, M., and Nakasuji, Y., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 251. Heitor, M. V., Taylor, A. M. K. P., and Whitelaw, J. H., J. Fluid Mech. 181:387 (1987). Goss, L. P., Trump, D. D., and Roquemore, W. M., Exp. Fluids 6:189 (1988). Hardalupas, Y., Tagawa, M., and Taylor, A. M. K. P., in Developments in Laser Techniques and Applications to Fluid Mechanics (R. J. Adrian et al., Eds.), SpringerVerlag, Berlin, 1996, p. 159. Libby, P. A., and Bray, K. N. C., AIAA J. 19:205 (1981). Bray, K. N. C., Libby, P. A., Masuya, G., and Moss, J. B., Combust. Sci. Technol. 25:127 (1981). Zhang, S., and Rutland, C. J., Combust. Flame 102:447 (1995). Veynante, D., and Poinsot, T., Center for Turbulence Research: Annual Research Briefs. NASA Ames Research Center & Stanford University, 1995, p. 273. Veynante, D., Trouve, A., Bray, K. N. C., and Mantel, T., J. Fluid Mech. 332:263 (1997). Tagawa, M., Nagaya, S., and Ohta, Y., Exp. Fluids 30:143 (2001). Humphrey, J. A. C., Whitelaw, J. H., and Yee, G., J. Fluid Mech. 103:443 (1981). Kim, W. J., and Patel, V. C., Trans. ASME: J. Fluids Engng. 116:45 (1994). Taylor, A. M. K. P., Whitelaw, J. H., and Yianneskis, M., Trans. ASME: J. Fluids Engng. 104:350 (1982). Bilger, R. W., in Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), Springer-Verlag, Berlin, 1980, p. 65.
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Stårner, S. H., and Bilger, R. W., Twenty-first Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, p. 1569. Driscoll, J. F., and Gulati, A., Combust. Flame 72:131 (1988). Jones, W. P., in Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), Academic Press, London, 1994, p. 309. Shepherd, I. G., and Moss, J. B., AIAA J. 20:566 (1982). Cheng, R. K., Combust. Sci. Technol. 41:109 (1984). Tagawa, M., and Ohta, Y., Combust. Flame 109:549 (1997). Tagawa, M., Shimoji, T., and Ohta, Y., Review Sci. Instrum. 69:3370 (1998). Collis, D. C., and Williams, M. J., J. Fluid Mech. 6:357 (1959). Bradley, D., Gaskell, P. H., Gu, X. J., Lawes, M., and Scott, M. J., Combust. Flame 115:515 (1998). Williams, F. A., Combustion Theory, The Benjamin/ Cummings Publishing Company, California, 1985, p. 1. Miyauchi, T., Tanahashi, M., Sasaki, K., and Ozeki, T., in Transport Phenomena in Combustion (S. H. Chan, Ed.), Taylor & Francis, New York, 1996, p. 1095. Libby, P. A., and Williams, F. A., in Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), SpringerVerlag, Berlin, 1980, p. 1.
Received 11 June 2001; revised 29 October 2001; accepted 28 November 2001
APPENDIX In the analysis of a multicomponent reacting system such as combustion [28], the energy equation can be expressed by enthalpy h.4 Thus, appropriate modeling of the turbulent enthalpy ⬃ flux u⬙k h⬙ is necessary for predicting combusting flows with turbulence models. In the following, we make clear the relationship between the ⬃ turbulent enthalpy flux u⬙k h⬙ and the turbulent ⬃ heat flux u⬙k t⬙ measured. The energy equation in terms of the enthalpy h can be expressed by [30]: 共 h兲, ⫹ 共 u kh兲, k ⫽
冋
冉
共h兲, k ⫹ D 共 j兲 cp
⫺
冊
册
h 共 y 兲, , , cp j j k k
(3)
4 In the direct numerical simulation of premixed turbulent combustion by Miyauchi et al. [29], the energy equation was expressed explicitly by temperature to avoid the iterative computation for obtaining temperature from enthalpy.
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where , and ,k denote the derivatives with respect to time and the spatial coordinate x k , respectively. Then, uk is a velocity component in the xk direction, and , h, , and cp are density, enthalpy, heat conductivity and the constantpressure specific heat of a mixture of gases, respectively. The enthalpy, mass fraction and diffusion coefficient of a chemical species j are denoted by hj, yj, and D(j), respectively, where j ⫽ 1 ⬃ n (n: total number of chemical species involved). The summation convention is applied to the indices k and j. It is noted that the time derivative of pressure, (p),, the effect of radiation, and the heat generation because of fluid viscosity are all neglected in Eq. 3. The following relation holds between enthalpy h and temperature T. h ⫽ y jh j ⫽ y j
冋冕
册
T
c pj dT ⫹ h 0j ,
T0
(4)
where T0 is a reference temperature (298.15 K), and cpj and h0j denote the constant-pressure specific heat and the standard heat of formation of the species j. Then, the mean value of the constant-pressure specific heat cpmj is defined by c pmj ⬅
冉 冕 冊冒 T
c pj dT
共T ⫺ T 0兲.
(5)
T0
(6)
Now, by modeling the turbulent enthalpy flux˜ u⬙ h⬙ using the gradient-diffusion hypothk
esis, we obtain
t ˜ ˜ 共h 兲, k, u ⬙kh⬙ ⫽ ⫺ c pm
(7)
where t is the eddy conductivity for heat. From the density-weighted (Favre) decomposition of ˜ ⫹ t⬙ and yj ⫽ ˜yj ⫹ y⬙j, h ˜ can be T and yj, T ⫽ T expressed by ˜ ⫽ c pmj 关y˜j共T ˜ ⫺ T 0兲 ⫹ ⬃ h y ⬙jt⬙ 兴 ⫹ ˜y jh 0j.
˜ 兲, k ⫹ c pmj 共⬃ y ⬙jt⬙ 兲, k其. ⫹ c pmj ˜y j 共T (9) Now, by rewriting the time-averaged value of the convection term uk h using uk ⫽ u ˜k ⫹ u⬙k ˜ ⫹ h⬙, we obtain and h ⫽ h ˜⫹ u kh ⫽ 共u ˜ kh
⬃ u ⬙kh⬙ 兲.
(10)
On the other hand, the substitution of Eq. 6 into uk h yields ⬃ ⬃ ˜ u kh ⫽ c pmj 关共u ˜ k˜y j ⫹ u ⬙ky ⬙j兲 共T ⫺ T 0兲⫹ ˜y j u ⬙kt⬙ ⬃ ⬃ ⫹u ˜ k y ⬙jt⬙ ⫹ u ⬙ky ⬙jt⬙兴⫹ 共u ˜ k˜y j ⬃ ⫹ u ⬙ky ⬙j兲 h 0j.
(11)
Hence, using Eqs. 8, 10, and 11, the turbulent enthalpy flux˜ u⬙k h⬙ can be decomposed into three sorts of turbulence quantities as follows: ˜ ˜ ˜ kh u ⬙kh⬙ ⫽ u kh/ ⫺ u ⬃ ˜ ⫺ T 0兲 ⫹ h 0j兴 u ⬙ky ⬙j ⫽ 关c pmj 共T ⬃ ⬃ ⫹ c pmj ˜y j u ⬙kt⬙ ⫹ c pmj u ⬙ky ⬙jt⬙ .
The substitution of Eq. 5 into Eq. 4 yields h ⫽ y j 关c pmj 共T ⫺ T 0兲 ⫹ h 0j兴.
t ⬃ ˜ ⫺ T 0兲 ⫹ h 0j兴共 ˜y j兲, k u ⬙kh⬙ ⫽ ⫺ 兵关c pmj 共T c pm
(8)
From the insertion of Eq. 8 into the righthand side of Eq. 7, we obtain a turbulence ⬃ model for u⬙k h⬙ :
(12) Applying the general gradient-diffusion models to the terms on the right-hand side of Eq. 12, we have ⬃ u ⬙ky ⬙j ⫽ ⫺D 共 j兲t 共 ˜y j兲, k
(13)
⬃ ˜ 兲, k u ⬙kt⬙ ⫽ ⫺共 t/ c pm兲 共T
(14)
⬃ ⬃ ⬃ ˜ /˜⑀ 兲 u u ⬙ky ⬙jt⬙ ⫽ ⫺c s 共k ⬙ku ⬙ᐉ 共 y ⬙jt⬙ 兲, ᐉ,
(15)
where D(j)t is the eddy diffusivity. Equation 15 is a conventional turbulence model for the triple ˜, and ˜⑀ denote a model product [21], and cs, k ⬃ constant, turbulence energy 共⫽ u⬙k u⬙k /2兲 and the ˜, respectively. dissipation rate of k Now, we can show a close relationship between the application of the gradient-diffusion
FLAME CHARACTERISTICS IN A RECTANGULAR DUCT ⬃ model. (Eq. 7) to u ⬙kh⬙ and the use of Eqs. 13-15. When substituting Eqs. 13-15 into Eq. 12 and comparing the result with Eq. 9, we may find that all the turbulence quantities on the right-hand side of Eq. 12 must obey the gradient-diffusion hypothesis in order for Eq. 7 to be an adequate model for the turbulent enthalpy flux. Because the contri⬃ bution of the triple product to u ⬙kh⬙ should
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be generally small, the behavior of the turbulent enthalpy flux will be dominated by ⬃ the turbulent heat and mass fluxes u ⬙k t⬙ ˜ and u ⬙ky ⬙j . Thus, it is very important to know whether or not the behaviors of these turbulence quantities can be adequately modeled with the gradient-diffusion hypothesis to justify the conventional turbulence modeling.