Stationary transverse rolls and U-rolls in limiting low Reynolds
number mixed convective air flow near the convective
threshold in a horizontal flat duct
T.C. Cheng, J.T. Lir, T.F. Lin
*Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30010, Taiwan Received 29 June 2000; received in revised form 6 July 2001
Abstract
Combined experimental flow visualization and temperature measurement are carried out in the present study to explore the buoyancy driven vortex flow patterns in a limiting low Reynolds number mixed convective air flow through a bottom heated horizontal flat duct. In Particular, attention is paid to the flow approaching the natural convection limit (Re¼ 0) for Re ¼ 1:0 and 2.0 with the Rayleigh number near the critical level for the onset of convection for 1200 6 Ra 6 4000. Results from the flow visualization have revealed two unfamiliar vortex flow patterns which were not seen in our earlier study [Int. J. Heat Mass Transfer 44 (4) (2001) 705]. One is characterized by the stable stationary transverse rolls in the duct entry and stable longitudinal rolls in the downstream. Another is in the form of U-rolls. The relations of these two patterns with those reported in the literature from analytical, numerical and experimental studies are discussed. Moreover, stable longitudinal rolls along with nonperiodic traversing waves, and mixed longitudinal and transverse rolls as well as irregular cells which appear in the higher Reynolds number for 3:0 6 Re 6 5:0 are also noted here. The temporal and spatial characteristics of the unfamiliar vortex flows are inspected in detail. In addition, the flow formation processes leading to the two unfamiliar vortex flow structures are also examined carefully. During the flow formation we noted merging of longitudinal and transverse rolls to form U-rolls, splitting of rolls into cells and the reverse process of cell integration into rolls, aside from the generation of the longitudinal and transverse rolls. Finally, a flow regime map is provided to delineate various vortex flow structures observed in this study and in the previous study (cf. the above-mentioned reference) driven by the slightly supercritical and subcritical buoyancies for 1:0 6 Re 6 5:0. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
Buoyancy driven vortex flow structure is known to be significantly affected by the Reynolds and Rayleigh numbers in a mixed convective flow through a bottom heated horizontal flat duct, as evident from various ex-perimental studies [1–7]. Depending on the level of the Reynolds number, longitudinal, transverse or mixed vortex rolls are frequently induced by the supercritical buoyancy. Recently, we [8] experimentally revealed two additional vortex flow patterns at slightly supercritical
and subcritical Rayleigh numbers for the relatively low Reynolds numbers with Re¼ 3:0, 4.0 and 5.0. More specifically, we observed a vortex flow characterized by steady longitudinal rolls near the duct sides along with nonperiodic moving transverse waves in the duct core. Another vortex flow is in the form of steady longitudinal rolls in the sidewall region, time periodic moving transverse rolls in the entry half of the duct core, and irregular cells in the exit half of the duct core. Besides, we also noted that even at these low Reynolds numbers steady longitudinal vortex flow is induced at slightly subcritical buoyancy. It is of interest to unveil the possible presence of the additional vortex flow structures for even lower Reynolds numbers for Re approaching zero. The understanding of this limiting low Reynolds number mixed convective vortex flow of gas is especially
International Journal of Heat and Mass Transfer 45 (2002) 1211–1227
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important in the growth of single crystal films from chemical vapor deposition [9].
In a mixed convective flow through a bottom heated horizontal flat duct the onset of the longitudinal vortex flow due to thermal instability was found to occur at the critical Rayleigh number RaL
c 1708 independent of the
Reynolds number of the flow and Prandtl number of the fluid [10–13]. Beyond this critical Rayleigh number, steady longitudinal rolls prevail and the roll diameter is nearly equal to the duct height. The corresponding spanwise temperature distribution has regular sinusoidal shape [10]. Kamotani and Ostrach [11,12] found that the regular sinusoidal temperature distributions were dis-torted and there were no stable vortex rolls for Re¼ 38 as Ra > 8000.
Various vortex structures driven by the buoyancy were delineated by a flow regime map including the flow with no vortex roll, steady and unsteady longitudinal rolls, transverse rolls and unsteady intermittent rolls [1– 5]. Moffat and Jensen [14,15] suggested that the buoy-ancy driven secondary flow structure was very sensitive to the duct aspect ratio. M€uuller et al. [16–18] solved the simplified modal amplitude equations for a low Rey-nolds number flow to predict the structures of longi-tudinal and transverse vortex flows.
The existence of the transverse thermoconvective rolls was proved by Luijkx and Platten [19] at very low Reynolds numbers. The critical Rayleigh number co-responding to the onset of the transverse rolls RaT c
was found to be a function of the aspect ratio and Prandtl number, and RaT
c increased with the Reynolds
number [19,20]. Beyond the critical Rayleigh number RaT
c, different vortex flow patterns may compete
de-pending on the Reynolds number. Steady and unsteady longitudinal rolls, time-periodic moving transverse rolls, and somewhat irregular snaking rolls were re-ported from our group [7]. Ouazzani et al. [3–5] refined the flow regime map to include the transverse rolls for air flow.
An interesting vortex flow pattern consisting of transverse rolls in the entry portion of the duct and longitudinal rolls in the downstream which prevailed at relatively low Reynolds numbers was predicted decades ago by a linear stability analysis from Cheng and Wu [21]. The pattern was later confirmed by the experimental flow observation [22] and numerical simulation [16,22–24]. The physical explanation for the appearance of this pattern was provided by M€uuller et al. [16].
Recently, combined flow visualization and tempera-ture measurement were carried out to explore the mixed convective air flow for 1 6 Re 6 50 and 1800 6 Ra 630,000 by Chang and Lin and their colleagues [6,7]. Their results revealed six vortex flow patterns : (1) stable longitudinal rolls, (2) unstable longitudinal rolls, (3) unstable longitudinal to transverse roll transi-tion, (4) mixed longitudinal/transverse rolls, (5) trans-verse rolls and (6) irregular rolls. Based on a linear stability analysis, Nicolas et al. [25] suggested that when the Rayleigh number exceeded certain critical value, the instabilities appeared in the form of three-dimensional horseshoe transversal rolls or longitudinal rolls.
A close examination of the above literature reveals that the buoyancy driven vortex flow in a mixed con-vective gas flow through a flat duct at a limiting low Reynolds number approaching zero remains poorly understood. To complement these early studies, an ex-periment including detailed flow visualization and tem-perature measurement is carried out here to investigate the temporal and spatial characteristics of the vortex flow structure for mixed convection of air in a horizontal plane channel at very low Reynolds number for Re 6 2:0. Note that at this low Reynolds number the buoyancy-to-inertia ratio Gr=Re2 is still relatively high even for a
subcritical buoyancy and it is reasonable to expect the existence of some vortex flow at the subcritical state [8,26]. Thus the Rayleigh number will be varied from 1200 to 4000.
Nomenclature
A aspect ratio (b=d) b; d channel width and height g gravitational acceleration Gr Grashof number (bgd3ðT
h TcÞ=m2)
Pr Prandtl number (m=a) Ra Rayleigh number (bgd3ðT
h TcÞ=am)
Rac critical Rayleigh number corresponding to the
onset of convection for an infinite layer Re Reynolds number (Wmd=m)
t time (s)
tp oscillation period (s)
T temperature
Tc; Th temperatures of the cold and hot plates
Tm mean temperature (ðThþ TcÞ=2)
W ; Wm velocity and average velocity components in z
direction
x, y, z dimensionless Cartesian coordinates scaled with d
a thermal diffusivity
b thermal expansion coefficient
h dimensionless temperature (ðT TmÞ=
ðTh TcÞ)
2. Experimental apparatus and procedures
The schematic diagram of the test apparatus which is modified slightly from our previous study [6] is shown in Fig. 1. In the following the apparatus is briefly described. More complete details on the ex-perimental system and procedures are given in our recent work [8]. The open-loop mixed convection ap-paratus consists of three parts: wind tunnel, test
sec-tion, and measuring bench for the velocity and temperature probes along with the data acquisition system. The test section is a rectangular duct of 240 mm wide and 300 mm long with 15 mm in height between the top cooled and bottom heated plates, providing an aspect ratio of A¼ 16. It should be noted that the duct height is reduced from 20 mm used in the previous study to 15 mm in this study. This narrower gap (d¼ 15 mm) could provide a better
Fig. 1. Schematic of test apparatus and the chosen coordinate system.
control of the temperature difference between the horizontal plates even at very low Rayleigh numbers to be covered in the present study. The bottom plate of the test section is made of a 20-mm-thick, high purity copper plate and is heated by dc power sup-plies. The top plate of the test section is made of a 3-mm -thick glass plate and a 2-3-mm-thick plexiglass plate with a gap width of 3 mm. This top plate is reinforced by copper alloy frames to keep it flat. Distilled water is provided from a tank and flows into this gap to cool the upper plate. The working fluid is air which is driven by a 7.5 hp air compressor and is sent into a 300-l and 100 psi high-pressure air tank. The air is first regulated by a pressure regulator and then is passed through a settling chamber, a contrac-tion nozzle, a developing seccontrac-tion and finally the test section. The developing section is 1660 mm in length, approximately 110 times of the duct height. This in-sures the flow being fully developed at the inlet of the test section for Re 6 50. An insulated outlet section of 160 mm long is added to the test section to reduce the effects of the disturbances from discharging the flow to the ambient surrounding of the open-loop wind tun-nel.
The volume flow rate of air is controlled and mea-sured by two Hasting HFC flow controllers designed especially for low volume flow rates, with accuracy better than 1%. A thermocouple probe, which is an OMEGA (model HYP-O) mini hypodermic extremely small T-type thermocouple (33 gauge) implanted in a 1 in. long stainless steel hypodermic needle, is used to measure the instantaneous temperature of the air flow in the duct.
Flow visualization is performed by injecting smoke tracer into the flow to observe the secondary flow structure. Specifically, the tiny incense smoke is in-jected into the main flow at some distance ahead of the settling chamber. By using a 1.5–2.5 mm plane light beam of an overhead projector with an adjust-able knife edge to illuminate the flow field containing these smoke particles, a sharp contrast could be ob-tained between the duct walls and the smoke.
Uncertainties in the Rayleigh number, Reynolds number and other independent parameters are calcu-lated according to the standard procedures established by Kline and McClintock [27]. The uncertainties of the thermophysical properties of the air are included in the analysis. The properties of the working fluid (air) are a¼ 0:22 ðcm2=sÞ, b ¼ 0:0034 (1/K), m ¼ 0:162
ðcm2=sÞ and Pr ¼ 0:74 at 30 °C and 1.0 bar. In
ad-dition, the uncertainties of the control unsteadiness and temperature nonuniformity are accounted for in the evaluation of the data uncertainty. The analysis shows that the uncertainties of temperature, volume flow rate, dimensions, Reynolds number and Rayleigh number measurements are estimated to be less than
0:15 °C, 1%, 0:05 mm, 2% and 5%, respec-tively.
3. Results and discussion
Selected results from the present study will be pre-sented in the following to illustrate the vortex flow structures in the limiting low Reynolds number mixed convective air flow for Re 6 2:0 subjected to the slightly supercritical and subcritical buoyancies for 1200 6 Ra 6 4000. Particular attention will be paid to some unfamiliar vortex flow patterns induced in the flow. How these vortex flow structures are formed in the flow will be examined in detail.
3.1. Vortex flow patterns at long time
To illustrate the vortex flow structures for various Rayleigh numbers at the limiting low Reynolds num-bers, the flow photos taken at the midheight of the duct (y¼ 1=2) from the top view at steady or statistically stable state are shown in the following. Firstly, the change of the vortex flow patterns is manifested in Fig. 2 for the Rayleigh number reducing from 4000 to 1200 at Re¼ 2:0. It has been known for some time that regular moving transverse rolls are normally formed in the flow at very low Re and high Gr=Re2 [3–6,19,24]. This
transverse vortex flow pattern can be seen for the Ray-leigh number of 2500 (Fig. 2(c)). Note that at this Ra the buoyancy-to-inertia ratio is so large that the moving transverse rolls are repeatedly induced at the duct inlet and are pushed slowly forward by the forced main flow. Raising the Rayleigh number from 2500 to 3000 causes the transverse rolls to grow in spanwise dimension and in strength. It is noted that the induced transverse rolls become somewhat bent and can merge with the neighbor rolls. Thus the vortex rolls are distorted to a certain degree and become unstable as traveling downstream (Fig. 2(b)). For a higher Ra of 4000 the distortion in the transverse rolls is more significant especially in the exit region of the duct (Fig. 2(a)). It is also noted that some irregular vortex flow in the form of connecting recircu-lating vortices is induced along the duct sides at the lower buoyancy with Ra¼ 2500 (Fig. 2(c)). For the lower Ra of 2000 the transverse rolls are weaker and do not extend to the duct sides. Instead, the longitudinal rolls are induced near the side walls of the duct. Besides, the transverse rolls disintegrate into a number of irreg-ular cells as they move to the exit half of the duct. Thus, at Ra¼ 2000 a mixed vortex flow including the longi-tudinal rolls near the duct sides, transverse rolls in the upstream core region and irregular cells in the down-stream core region is induced in the duct (Fig. 2(d)). At an even lower Ra of 1750 the transverse rolls and ir-regular cells completely disappear in the duct core and
more longitudinal rolls are induced near the existing longitudinal rolls (Fig. 2(e)). Hence a steady longitudinal vortex flow prevails in the duct. For a further reduction of Ra to the subcritical level (Ra¼ 1650, 1500 and 1200),
only a few longitudinal rolls appear near the duct sides (Figs. 2(f)–(h)). It is of interest to observe that some weakly transverse waves are generated nonperiodically in the core region for the cases with Ra¼ 1650 and 1500.
Fig. 2. Top view flow photos at steady or statistical state for Re¼ 2:0 and (a) Ra ¼ 4000, (b) Ra ¼ 3000, (c) Ra ¼ 2500, (d) Ra ¼ 2000, (e) Ra¼ 1750, (f) Ra ¼ 1650, (g) Ra ¼ 1500 and (h) Ra ¼ 1200.
These transverse waves are pushed by the main forced flow and move downstream. As the buoyancy is further decreased (Ra¼ 1200) these traversing waves disappear and no other secondary flow is induced in the duct core. This appearance of the vortex flow at subcritical buoy-ancies in the sidewall region is attributed to the fact that near the duct sides the forced main flow is at much lower speed than that in rest of the duct and the cross plane secondary flow can be initiated by the slightly subcritical buoyancy. The vortex flow driven by the subcritical buoyancy was also shown in a numerical simulation to be significant by Ouazzani and Rosenberger [28] in horizontal chemical vapor deposition processes.
Some unfamiliar vortex flow structures are revealed in Fig. 3 by presenting the top view flow photos for various Ra at the lower Reynolds number with Re¼ 1:0. The results for Re¼ 1:0 show that at the high Rayleigh numbers of 4000 and 3000 the buoyancy-to-inertia ratios are so large that highly deformed moving trans-verse rolls dominate in the duct (Figs. 3(a) and (b)). Obviously, the irregularity in the roll pattern is more severe for the case with a higher buoyancy-to-inertia ratio. The transient temperature measurement indicated that the flow oscillated chaotically in time. The details on the flow regimes observed in this study will be dis-cussed later. It is important to note from Fig. 3(c) that a drastic change in the vortex flow pattern occurs for a reduction of Ra from 3000 to 2500. Specifically, at Ra¼ 2500 the vortex flow is steady and is characterized by a stationary transverse roll at the inlet of the duct, followed immediately by 16 steady longitudinal rolls. All the longitudinal rolls have nearly the same diameter. This unique vortex pattern was not observed in our previous study [6]. However, the presence of this pattern was shown by the stability analysis and numerical and experimental explorations [16,21–24], from various workers as indicated in the last section. The inlet transverse roll is essentially a steady return flow induced by the sudden heating of the flow as it moves from the upstream unheated region into the heated section of the duct. As the buoyancy is lowered further to Ra¼ 2000, the stationary transverse roll merges with the longitu-dinal rolls adjacent to the duct sides to form a U-roll (Fig. 3(d)). Meanwhile, the beginning portion of the longitudinal rolls near the duct inlet can merge together to form another transverse roll and this transverse roll then merges with the other longitudinal rolls to form a smaller U-roll. Thus, we have another unfamiliar vortex flow which consists of U-rolls and longitudinal rolls. This vortex flow is periodic in time. The U-rolls were reported from a numerical simulation by Spall [29,30] in a lower aspect ratio duct. For even lower Ra of 1750, 1650 and 1500 the stationary transverse rolls at the duct inlet do not merge with the longitudinal rolls. Thus, we have inlet stationary transverse rolls followed by the steady longitudinal rolls (Figs. 3(e)–(g)), similar to that
shown in Fig. 3(c) for Ra¼ 2500. But for the lower Rað¼ 1750–1500Þ less longitudinal rolls are induced in the duct and more than one stationary transverse rolls can exist in the duct entry. It is also evidenced from the side view flow photos in Fig. 4 for selected vertical planes that the transverse rolls in the duct inlet are es-sentially stationary for Ra¼ 2500, 1750 and 1650. But at Ra¼ 2000, the transverse rolls travel a short distance into the test section and merge with the longitudinal rolls in the downstream to form U-rolls (Fig. 4(b)). The results in Fig. 4 also suggest that at a lower Ra the inlet transverse rolls are shorter, smaller and weaker. Besides, transverse rolls also appear near the duct exit for Ra¼ 1750 and 1650 (Figs. 4(c) and (d)). Note that the stationary transverse rolls induced near the duct entry and exit were observed and examined in detail by Visser et al. [31] and Ingle and Mountziaris [32] even at highly subcritical Rayleigh numbers. Besides, these stationary transverse rolls often appear in various metal–organic chemical vapor deposition processes [28,33,34]. At the lowest buoyancy tested here with Ra¼ 1200 nonperiodic traversing transverse waves appear in nearly the entire duct except in the side wall region where a few steady longitudinal rolls prevail (Fig. 3(h)). It should be pointed out here that repeatability tests of the unfamiliar vortex flow patterns examined above have been carried out to ascertain their existence.
3.2. Periodical change in vortex flow structures
For the two unfamiliar vortex flow patterns discussed above, the flow consisting of the inlet stationary trans-verse rolls and downstream longitudinal rolls is steady as the initial transient dies out. On the other hand the flow containing U-rolls is periodic in time. The periodic evolution of the U-rolls is examined in the following. Fig. 5 shows the change in the vortex flow comprising the U-rolls in a typical periodic cycle for the case with Re¼ 1:0 and Ra ¼ 2000. The period of the vortex flow tp
is 203 s. Here the dimensional time t denotes certain time instant in the statistical state for that case. Note that at time t a U-roll is formed from the merging of the transverse roll at the duct inlet with the two longitudinal rolls adjacent to the duct sides (Fig. 5(a)). Meanwhile, the upstream ends of the two neighboring longitudinal rolls merge together to form seven slender U-rolls. It is of interest to note that as time proceeds, the transverse portion of the large U-roll slowly moves downstream pushing the slender U-rolls forward and in the mean time a new transverse roll is generated at the duct inlet (Fig. 5(b)). Then, this new transverse roll merges grad-ually with the two slender U-rolls nearest to the duct sides (Figs. 5 (b) and (c)). As the process continues, more and more moving transverse rolls are induced and six large U-rolls are formed in the duct at time tþ 3=8tp
trans-verse rolls and the slender U-rolls are somewhat weak and can be broken away to initiate a reverse process for the second half of the periodic cycle for t P tþ 1=2tp.
The vortex flow then becomes that shown in Fig. 5(e)
including an outermost U-roll, some slender U-rolls, and some short and weak transverse rolls. For a further increase in time the weak transverse rolls in the duct core degenerate and break into cells. Later these cells merge
Fig. 3. Top view flow photos at steady or statistical state for Re¼ 1:0 and (a) Ra ¼ 4000, (b) Ra ¼ 3000, (c) Ra ¼ 2500, (d) Ra ¼ 2000, (e) Ra¼ 1750, (f) Ra ¼ 1650, (g) Ra ¼ 1500 and (h) Ra ¼ 1200.
with the downstream longitudinal rolls (Figs. 5(f) and (g)). At increasing time, most U-rolls gradually disinte-grate into longitudinal rolls (Fig. 5(h)). Finally at tþ tp,
the flow returns to its original state at time t to have only one big U-roll in the duct. To further unravel the structure of the time periodic U-rolls, the corresponding end view flow photos at selected cross sections of the duct are displayed in Fig. 6 for selected time instants. The results suggest that in the entry region of the duct for z¼ 1 and 3.36 the vortices change drastically with time implying the formation and disintegration of the U-rolls in this region. But in the downstream where the
longitudinal rolls prevail the change in the vortices is rather mild.
The temporal change of the temperature in the flow is rather significant for the regular and deformed trans-verse rolls and is shown in Fig. 7 by presenting the time records of the air temperature at selected locations for some cases. The results in Fig. 7(a) for Re¼ 2:0 and Ra¼ 2500 indicate that in the region dominated by the regular transverse rolls the flow is time periodic as our previous study [8]. Moreover, the transverse vortex flow oscillates at the same frequency (tp¼ 11:1 s) and
am-plitude in the duct core where the fully developed
Fig. 4. Side view flow photos at selected vertical planes for Re¼ 1:0 and (a) Ra ¼ 2500, (b) Ra ¼ 2000, (c) Ra ¼ 1750 and (d) Ra¼ 1650.
transverse rolls prevail. It is of interest to notice that even in the region slightly upstream of the test section the flow oscillates weakly at the same frequency as that in the test section. At higher buoyancy-to-inertia ratios the transverse rolls become larger and can merge
to-gether, and the vortex flow becomes somewhat irregular (Figs. 7(b) and (c)). Due to the presence of the irregular vortex rolls in the downstream some irregularity in the temperature oscillation is noted. This irregularity is more severe by further increasing the
buoyancy-to-Fig. 5. Top view flow photos for the U-rolls in a typical periodic cycle for Re¼ 1:0 and Ra ¼ 2000 at time ¼ (a) t, (b) t þ 1=8tp,
(c) tþ 1=4tp, (d) tþ 3=8tp, (e) tþ 1=2tp, (f) tþ 5=8tp, (g) tþ 3=4tpand (h) tþ 7=8tp(tp¼ 203 s).
inertia ratio to the case with Re¼ 1:0 and Ra ¼ 4000 (Fig. 7(d)).
The time periodic evolution of the vortex flow pat-tern consisting of the steady longitudinal rolls in the side wall region of the duct, moving transverse rolls in the
duct entry and the irregular recirculating cells in the downstream core region of the duct for the case with Re¼ 2:0 and Ra ¼ 2000 resembles that for the higher Reynolds number with Re¼ 3:0 discussed in the pre-vious study [8] and the details of this evolution are not
Fig. 6. Top view flow photo for the U-rolls and the corresponding end view flow photos in a typical periodic cycle for Re¼ 1:0 and Ra¼ 2000 at (a) z ¼ 1 (tp¼ 38 s), (b) z ¼ 3:36 ðtp¼ 203 s) and (c) z ¼ 10:09 (tp¼ 203 s).
given here. Moreover, the observed time periodic mov-ing transverse vortex flow at Re¼ 2:0 and Ra ¼ 2500 is also similar to that at higher Re [6,8].
3.3. Formation of unfamiliar vortex flow patterns
In the mixed convective air flow through a horizontal flat duct heated from below two unfamiliar vortex flow patterns were observed and were examined above for Ra near the critical threshold at Re¼ 1:0. It is important in the fundamental study of heat transfer and fluid dy-namics to unveil the processes through which these vortex structures are formed. In the study of the vortex flow formation the experiment is started by setting the Reynolds number at 20.0 and the Rayleigh number at the value for the case to be investigated so that the buoyancy-to-inertia ratio is rather low for a sufficient period of time. Thus, the initial flow (t < 0) established in the duct is forced convection dominated. Then, at
time t¼ 0 the Reynolds number of the flow is lowered quickly to the level for the specific case to be examined and maintained at this level thereafter for t > 0. Note that due to the flow inertia, normally it takes about 10– 20 s for the Reynolds number to be reduced to the re-quired level.
The formation of the interesting vortex flow pattern consisting of stationary transverse rolls in the entry re-gion of the duct and stable longitudinal rolls in the downstream is presented in Fig. 8 for Re¼ 1:0 and Ra¼ 1750. Note that this unique vortex pattern is only observed at the lowest Reynolds number tested here at Re¼ 1:0 (Figs. 2 and 3). The results in Fig. 8 indicate that immediately after the reduction of the Reynolds number from 20.0 to 1.0 transverse rolls are repeatedly generated at the duct inlet and longitudinal rolls are successively induced near the duct sides. Note that at this low Reynolds number the transverse rolls travel downstream at a very low speed and they are seriously
Fig. 7. Side view flow photos at statistical state and the corresponding time records of air temperature at selected locations on the line y¼ 1=2 and x ¼ 0 for (a) Re ¼ 2:0, Ra ¼ 2500 (tp¼ 11:1 s), (b) Re ¼ 2:0, Ra ¼ 3000 (tp¼ 11:1 s), (c) Re ¼ 2:0, Ra ¼ 4000 (tp¼ 12:1 s)
and (d) Re¼ 1:0, Ra ¼ 4000 (tp¼ 21:9 s).
blocked by the newly induced longitudinal rolls away from the duct sides, as evident from the flow photos in Figs. 8(b) and (c). In fact, the transverse rolls can only travel a short distance before they are completely
stop-ped (Fig. 8(d)). Then the front transverse rolls weaken gradually and later disappear (Fig. 8(e)). The remaining transverse rolls near the duct inlet, however, stay sta-tionary in the duct entry and no new transverse rolls are
Fig. 8. Top view flow photos showing the formation of stationary transverse rolls in the duct entry and downstream longitudinal rolls by lowering Re from 20.0 to 1.0 in 14.2 s for Ra¼ 1750 at time t ¼ (a) 0 s, (b) 19 s, (c) 28 s, (d) 37 s, (e) 77 s, (f) 117 s, (g) 194 s, (h) 346 s, (i) 446 s, (j) 615 s, (k) 807 s and (l) 1079 s.
generated thereafter (Figs. 8 (f) and (g)). As time pro-ceeds, the stationary transverse rolls get stronger and more longitudinal rolls are induced (Figs. 8(h)–(k)). Fi-nally at t > 1050 s, a vortex flow pattern characterized by the inlet stationary transverse rolls along with the downstream stable longitudinal rolls is formed (Fig. 8(l)).
The above vortex flow structure containing the inlet stationary transverse rolls and downstream stable longitudinal rolls also exists at a higher buoyancy for Re¼ 1:0. Fig. 9 shows the formation of such vortex flow driven at Ra¼ 2500. At this higher Ra the transverse and longitudinal rolls are induced in a faster pace fol-lowing the lowering of the Reynolds number and the rolls are stronger (Fig. 9(b)). It is of interest to note that near the duct exit transverse rolls are also induced at this higher Ra. Later, the inlet and exit transverse rolls merge with the longitudinal rolls to form rectangular rolls (Figs. 9(c)–(e)). Meanwhile, the transverse rolls in the duct entry slowly move downstream and hence push the rectangular rolls forward. Besides, new transverse roll is repeatedly generated at the duct inlet. Note that the downstream end of the rectangular rolls leaves the duct at a certain time instant and we observe slightly de-formed U-rolls in the duct (Fig. 9(f)). As time proceeds, the transverse rolls weaken and the U-rolls become more deformed (Fig. 9(g)). Moreover, the outer U-rolls near the duct inlet and duct sides split into slightly deformed transverse and longitudinal rolls. Then, the deformed transverse rolls disintegrate into recirculating cells. These cells later merge together to form short longi-tudinal rolls (Figs. 9 (h) and (i)). The longilongi-tudinal rolls gradually strengthen up and extend towards the duct exit to merge with the U-rolls. This process continues and more and more longitudinal rolls are formed (Figs. 9(i)–(k)). But only one transverse roll exists at the duct inlet. The deformed longitudinal rolls take certain period of time to become straight, and finally at t >510 s a vortex flow pattern comprising of an inlet stationary roll and downstream stable longitudinal rolls is reached.
In the vortex flow evolution leading to the U-rolls, moving transverse rolls are also repeatedly generated in the duct entry and longitudinal rolls are induced near the duct sides in the initial transient (Figs. 10(b) and (c)). Later, the transverse and longitudinal rolls merge to-gether to form a few U-rolls (Figs. 10(d) and (e)). These U-rolls are distorted to a certain degree and the roll distortion increases with time. Note that at t¼ 149 s the rolls become rather irregular. But a stationary, outer-most U-roll with its legs along the duct sides is clearly seen (Fig. 10(f)). Besides, a regular transverse roll is seen near the duct inlet. Then, it is of interest to observe the appearance of short longitudinal rolls in the entry half of the duct (Fig. 10(g)). As the process continues, these longitudinal rolls grow slowly in the downstream
di-rection. Meanwhile, the transverse roll and longitudinal rolls merge together to form a U-roll (Fig. 10(h)). Then, the transverse portion of the U-rolls move slowly downstream and a new transverse roll is generated at the duct inlet. This newly formed transverse roll and the transverse portions of the U-rolls merge with the longitudinal rolls to form U-rolls. In this process one more U-roll appears (Figs. 10 (i) and (j)). Note that the U-rolls away from the duct entry are rather weak and later disintegrate into cells (Figs. 10(i) and (j)). These cells then merge together to form longitudinal rolls (Fig. 10(k)). Note that in the region between the U-rolls there appear several longitudinal rolls. Finally, at t > 680 s a vortex flow pattern characterized by a few U-rolls with several longitudinal rolls contained in the space between the U-rolls is formed.
3.4. Flow regime map
Based on the present data and the data obtained in the previous study [8], various vortex flow patterns in-duced in the low Reynolds number mixed convective air flow in a bottom heated horizontal flat duct are sum-marized in a flow regime map given in Fig. 11. Also indicated in the map is the condition around which the vortex flows consisting of stable longitudinal rolls near the duct sides and nonperiodic traversing transverse waves in the duct core. This condition can be roughly represented by the equation
RaNWc 1750 540=Re2
: ð1Þ
Moreover, the boundary between the regular transverse vortex flow and the mixed vortex flow which is charac-terized by the stable longitudinal rolls near the duct sides and the time periodic moving transverse rolls in the duct core can be approximately expressed as
RaT
c 2200 þ 1:7 Re
4: ð2Þ
Finally, the transition between the regular transverse rolls and deformed transverse rolls occurs at
RaITc 2700 þ 0:2 Re 9
: ð3Þ
For comparison purpose the vortex flow patterns for the limiting case of the natural convective flow in the test section, i. e., for Re¼ 0, are also indicated in the map. In the natural convection test the inlet and exit of the test section are completely closed by placing plexiglass plates at these locations. Note that for Re¼ 0 the entire test section, which is essentially a horizontal rectangular enclosure, is filled with the sta-tionary transverse rolls for 2000 6 Ra 6 3000. The rolls are parallel with the short sides of the enclosure [26]. For Ra¼ 4000 deformed transverse rolls prevail. But at the lower buoyancy for Ra 6 1750 rectangular rolls dominate in the enclosure. In broad sense these vortex
flow patterns for Re¼ 0 are somewhat related to those at Re¼ 1:0. The details on these natural convective vortex flow patterns were examined in an earlier study [35].
4. Concluding remarks
A study combining experimental flow visualization and temperature measurement has been carried out here
Fig. 9. Top view flow photos showing the formation of stationary transverse rolls in the duct entry and downstream longitudinal rolls by lowering Re from 20.0 to 1.0 in 14.5 s for Ra¼ 2500 at time t ¼ (a) 0 s, (b) 6 s, (c) 8 s, (d) 14 s, (e) 20 s, (f) 39 s, (g) 58 s, (h) 152 s, (i) 178 s, (j) 247 s, (k) 296 s and (l) 519 s.
to explore the buoyancy driven vortex flow structures in a mixed convective air flow in a bottom heated hori-zontal flat duct at a very low Reynolds number ap-proaching the natural convection limit for the Rayleigh number around the critical level for the onset of
con-vection, including the subcritical and supercritical states. Results from the present study have revealed some in-teresting but unfamiliar vortex flow structures. In par-ticular, we identify two unfamiliar structures, namely, the vortex flow comprising stationary transverse rolls in
Fig. 10. Top view flow photos showing the formation of U-rolls by lowering Re from 20.0 to 1.0 in 14.8 s for Ra¼ 2000 at time t ¼ (a) 0 s, (b) 6 s, (c) 28 s, (d) 43 s, (e) 69 s, (f) 149 s, (g) 201 s, (h) 275 s, (i) 496 s, (j) 612 s, (k) 620 s and (l) 716 s.
the duct entry and stable longitudinal rolls in the downstream, and the vortex flow containing U-rolls.
The formation processes leading to these two vortex flow patterns observed in the present experiment are also examined in detail. The results disclose many complicate processes during the vortex flow formation, including the generation of the longitudinal and transverse rolls, merging of longitudinal and transverse rolls to form U-rolls, splitting of rolls into cells and the reverse process of integrating cells into rolls. Besides, a flow regime map is given to delineate various vortex flow patterns in the limiting low Reynolds number mixed convective flow.
Acknowledgements
The financial support of this study by the engineering division of National Science Council of Taiwan, ROC through the contract NSC83 0404 E009 054 is greatly appreciated.
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Eq. (3)
Eq. (2)
Eq. (1) : T rolls, :Deformed T rolls, : L rolls, : Mixed L & T rolls, : U-rolls, : Rectangular rolls, : Nonperiodic Transverse waves & L rolls, : Mixed L & T rolls & downstream irregular cells,
: Inlet stationary T rolls & downstream longitudinal rolls.
0 1 2 3 4 5 1000 1500 2000 2500 3000 3500 4000 Ra Re
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