Vortex ¯ow patterns near critical state for onset of
convection in air ¯ow through a bottom heated horizontal
¯at duct
J.T. Lir, M.Y. Chang, Tsing-Fa Lin*
Department of Mechanical Engineering, National Chiao Tung University, 1001 Ra Hsueh Road, Hsinchu 30049, Taiwan Received 31 January 2000; received in revised form 7 April 2000
Abstract
In this paper, an experiment is carried out to investigate the buoyancy driven vortex ¯ow at slightly subcritical and supercritical Rayleigh numbers in a mixed convective air ¯ow through a bottom heated horizontal ¯at duct. Particular attention is paid to the ¯ow at a very low Reynolds number for 3:0RReR5:0: Results from the ¯ow visualization have revealed two new vortex ¯ow patterns in addition to those often seen in literature such as longitudinal rolls (L rolls), moving transverse rolls (T rolls), and mixed longitudinal/transverse rolls (M rolls). The newly observed vortex ¯ow patterns include the stable longitudinal rolls near the duct sides along with nonperiodic traversing transverse waves in the duct core, and the mixed longitudinal and transverse rolls as well as irregular cells. Moreover, steady longitudinal rolls and nonperiodic traversing transverse waves are noted even at subcritical Rayleigh numbers. A ¯ow regime map delineating various vortex ¯ow patterns is given. Temporal characteristics of the ¯ow are also inspected. Furthermore, a correlation is given to estimate the local onset locations of the longitudinal rolls. Meanwhile, the oscillation frequency and convection speeds of the transverse rolls are correlated from the present data. Finally, many complicate processes during the vortex ¯ow formation are noted, such as the generation of the L and T rolls and transverse waves, splitting of rolls into cells and the reverse process of cell integration into rolls, aside from the moving and bending of T rolls. 7 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Vortex ¯ow; Near critical mixed convection; Flat duct
1. Introduction
Considerable research has been carried out recently to study the buoyancy driven vortex ¯ow in mixed convection of gas through a horizontal
¯at duct due to its importance in various techno-logical processes such as cooling of microelectronic equipments [1], design of compact heat exchangers [2] and thin crystal ®lm growth from chemical vapor deposition (CVD) [3±5]. At moderately super-critical Rayleigh numbers various vortex ¯ow struc-tures such as longitudinal rolls, transverse rolls and mixed longitudinal/transverse rolls have been reported in the literature. However, in the limiting Rayleigh number very close to the onset of
convec-0017-9310/01/$ - see front matter 7 2001 Elsevier Science Ltd. All rights reserved. PII: S0017-9310(00)00137-X
* Corresponding author. Tel.: +886-35712121-55118; fax: +886-35726440.
tion for Ra11708, the vortex ¯ow can be very dierent from those at high Ra and remains largely unexplored. Some unusual vortex ¯ow patterns may appear, as discussed by Koschmieder [6] on natural convection in a shallow cavity.
The critical Rayleigh number RaL
c for the onset
of longitudinal vortex rolls in a mixed convective ¯ow through a bottom heated horizontal ¯at duct was found to be 1708 [7±12]. It is well known that RaL
c is unaected by the Reynolds number Re. But
the critical Rayleigh number for the onset of the transverse rolls RaT
c increases with the Reynolds
number Re [13,14]. Slightly above the critical Ray-leigh number RaL
c, steady longitudinal rolls prevail
for Rer9 and the roll diameter is nearly equal to the duct height [8]. As the Rayleigh number is well above RaL
c, there is no stable vortex rolls [9,10].
Flow regime maps in terms of the Reynolds num-ber versus Rayleigh numnum-ber were proposed by var-ious workers to delineate the ¯ow with no vortex roll, steady and unsteady longitudinal, transverse and intermittent rolls [15±19]. Kamotani et al. [12] indicated that in the thermal entrance region the heat transfer rate was aected not only by the Ray-leigh number but also by the buoyancy-to-inertia ratio Gr/Re2.
Luijkx and Platten [13] proved the existence of the transverse thermoconvective rolls in mixed con-vection of silicone oil Pr1450 through a bottom heated horizontal plane channel at a very low Rey-nolds number, 0:001RRaR0:01: The critical Ray-leigh number for the onset of the transverse rolls RaT
c was found to be a function of the aspect ratio
and Prandtl number. Ouazzani et al. [17±19] re®ned the ¯ow regime maps to include the transverse rolls
for air ¯ow with 1 < Re < 9 and
12,000RRaR20,000: They also found a particular
¯ow regime that appeared approximately between 1920 < Ra < 2082 and 0:45 < Re < 0:75, in which only irregular and intermittent ¯ow structure was observed. Recently, Chang and Lin and their col-leagues [20±24] carried out detailed ¯ow visualiza-tion and temperature measurement to explore the
mixed convective air ¯ow for ReR50 and
30,000rRar1800: They revealed six vortex ¯ow patterns: (1) stable longitudinal rolls, (2) unstable longitudinal rolls, (3) unstable longitudinal to trans-verse roll transition, (4) mixed longitudinal/trans-verse rolls, (5) translongitudinal/trans-verse rolls and (6) irregular rolls.
In an experimental study Kamotani et al. [12] indicated that the steady longitudinal vortex ¯ow became unstable once the fully developed vortex rolls occupied the entire test section in a long plane channel. For a larger duct aspect ratio the ¯ow is more irregular. Moat and Jensen [25,26] suggested that the bouyancy driven secondary ¯ow structure was very sensitive to the aspect ratio of the duct.
Similar investigations with the bottom plate heated by a uniform heat ¯ux were conducted by Incropera et al. [27±30]. Four ¯ow regimes with progressing complexity were identi®ed along the channel, beginning with laminar forced convection near the duct inlet, followed by laminar mixed con-vection, transitional mixed convection and turbulent free convection.
The above literature review clearly indicates that the buoyancy driven vortex ¯ow in a mixed convective gas ¯ow through a ¯at duct at a low Rayleigh number just exceeding the critical level remains largely unexplored. In the present study, an experiment combining the ¯ow visualization and temperature measurement is car-ried out to investigate the temporal and spatial charac-teristics of the vortex ¯ow structure induced in mixed Nomenclature
A aspect ratio, b/d
b, d channel width and height
f, F dimensional and dimensionless frequencies, F f= a=d2
g gravitational acceleration Gr Grashof number, bgd3 T
hÿ Tc=n2
Pr Prandtl number, n=a
Ra Rayleigh number, bgd3 T
hÿ Tc=an
Rac critical Rayleigh number corresponding to
onset of convection
Raz local Rayleigh number, bgz3 Thÿ Tc=an
Re Reynolds number,Wmd
n
t time (s)
tp oscillation period (s)
T temperature
Tc, Th temperatures of cold and hot plates
Tm mean temperature, Th Tc)/2
W, Wm velocity and average velocity components
in z direction
Wr wave speed of the transverse rolls
x, y, z dimensionless cartesian coordinates scaled with d
a thermal diusivity
b thermal expansion coecient y dimensionless temperature, TÿTm
ThÿTc
convection of air through a bottom heated, horizontal ¯at duct at very low Reynolds number with Re varying from 3 to 5. This low Reynolds number ¯ow is typical for CVD processes [5]. Note that at this low Reynolds number the bouyancy-to-inertia ratio Gr/Re2 is still relatively high even for the slightly subcritical Rayleigh number and it is reasonable to expect the existance of some vortex ¯ow at the subcritical state [6]. Thus, the Rayleigh number will be varied from 1200 to 4000. 2. Experimental apparatus and procedures
2.1. Experimental apparatus
The schematic diagram of the test apparatus which is modi®ed slightly from our previous study [20,23,24] is shown in Fig. 1. The open-loop mixed convection apparatus consists of three parts: wind tunnel, test sec-tion, and measuring bench for the velocity and tem-perature probes along with the data acquisition system. The test section is a rectangular duct 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 A 16: It should be noted that the duct height is reduced from 20 mm used in our pre-vious study to 15 mm in this study. This narrower gap d 15 mm) could provide a better control of the tem-perature dierence between the horizontal plates even at very low Rayleigh numbers to be covered in the pre-sent study, since at the same low Rayleigh number the temperature dierence DT is much larger for d 15 mm in view of the fact that RaAd3 DT: Thus the
ex-perimental uncertainty in maintaining a constant DT exhibits much smaller eect on the measured data. The bottom plate of the test section is made of a 20 mm thickness, high purity copper plate and is heated by DC power supplies. In order to insure uniform bottom plate temperature, the heaters attached onto the out-side surface of the bottom copper plate are divided into 10 segments in the main ¯ow direction and each heater is independently controlled by a GW GPC 3030 D laboratory power supply. In order to reduce the edge eect of the test section, the width of the bottom copper plate is 40 mm larger than the bottom of test section. The top plate of the test section is made of 3 mm thick glass plate and 2 mm thick plexiglass plate with a gap width of 3 mm. This top plate is reinforced by copper alloy frames to keep it ¯at. Distilled water is provided from a tank and ¯ows into this gap to cool the upper plate.The distilled water is maintained at a constant temperature by a constant temperature circu-lation unit that consists of a cooler, a heater and a 200 l distilled water reservoir. This cooling unit can control the temperature of the distilled water within 20.18C. The water ¯ow rate is adjusted carefully to keep the
temperature dierence over the top plate within 20.18C. The water head is also suitably adjusted to minimize any possible deformation in the glass plate.
The working ¯uid 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 ®rst regulated by a pressure regulator and then is passed through a settling chamber, a contraction nozzle, a developing section and ®nally the test section. The purpose of forcing the air through the settling chamber is to reduce the air turbulence by installing a diuser buer section, two ®ne mesh screens, a honeycomb section and ®nally four ®ne mesh screens in the settling chamber. The air turbulence and ¯ow separation are further suppressed by the contraction nozzle section with a contraction ratio of 20:1, which provides a nearly uniform velocity at the inlet of the developing section.
The developing section is 1660 mm in length, ap-proximately 110 times of the duct height. This insures the ¯ow being fully developed at the inlet of the test section for ReR50: An insulated outlet section of 160 mm length is added to the test section to reduce the eects of the disturbances from discharging the ¯ow to the ambient surrounding of the open-loop wind tunnel. The developing and outlet sections are both made of 5 mm thick plexiglass plates, whereas the settling chamber and contraction nozzle are made of stainless steel plates. The entire loop is thermally insulated with a superlon insulator of 20 mm thickness and is bounded on a rigid supporting frame.
The volume ¯ow rate of air is controlled and measured by two Hasting HFC ¯ow controllers designed especially for low volume ¯ow rates, with ac-curacy better than 1%. In particular, the low air ¯ow rate is controlled precisely by the thermal mass ¯ow sensing technique [31]. These two ¯ow controllers indi-vidually operate in the ranges of 0±15 and 0±3 l/min, and they are calibrated by a Brooks bell prover with an accuracy of 0.2%. The operating condition of the ¯owmeter in the actual experiment is adjusted to a condition similar to that of the calibration stand.
The temperature of the test section is monitored by 13 calibrated and electrically insulated copper±constan-tan (T-type) thermocouples embedded in the bottom plate and six T-type thermocouples stuck to the inside surface of the top plate. The temperature of both plates could be maintained at nearly uniform and con-stant values with the deviations ranging from 20.05 to 20.128C. Additional thermocouples are used to measure the temperature of the inlet and outlet air. For each experiment, the top plate temperature is kept at the same values as that of the inlet air ¯ow for the purpose of eliminating the formation of a thermal boundary layer on the top wall. The thermcouple sig-nals are recorded by the Computer Products RTP 743 data acquisition system with a resolution of 20.058C.
A thermocouple probe, which is an Omega (model HYP-O) mini hypodermic extremely small T-type ther-mocouple (33 gauge) implanted in a 1 in. long stainless steel hypodermic needle, is used to measure the instan-taneous temperature of the air ¯ow in the duct. The probe is supported by a three-way traversing stand, which is inserted into the ¯ow from the exit end of the channel. In addition, the velocity pro®le is measured by a hot-wire probe. For calibrating the hot wire, the pipe ¯ow method that the probe is placed in the center of a fully developed laminar pipe ¯ow is used. The total volume ¯ow rate is measured and the pipe center velocity is calculated from the parabolic distribution. These data are recorded by an HP data acquisition system (Hewlett-Packard VXI series-E1411B multi-meter and E-1347A multiplexers).
2.2. Analysis of temperature oscillation
In order to unravel the unsteady characteristics of the vortex ¯ow, the transient temperature oscillations are obtained at selected detection points. The sampling rate of the data channel in the data acquisition system is set at 0.08 s per scan which is much shorter than the period of the ¯ow oscillation in the low Reynolds number mixed convective ¯ow considered here. 2.3. Preliminary investigation of ¯ow ®eld and experimental procedures
In order to con®rm the fully developed laminar ¯ow at the entrance of the test section, the main forced ¯ow distribution is measured by a hot wire probe which is operated by a constant temperature anem-ometer (DANTEC Probe Type 55P01 with 56C17 CTA Bridge). The measured data indicated that at the inlet of the test section the velocity pro®le is fully developed and is in good agreement with the analytical results given by Shah and London [32]. Besides, the turbulence intensities in the ¯ow were found to be less than 1%, implying that the eects of the free stream turbulence on the mixed convective ¯ow characteristics were moderate.
Flow visualization is performed by injecting smoke tracer into the ¯ow to observe the secondary ¯ow structure. Speci®cally, the tiny incense smoke is injected into the main ¯ow 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 adjustable knife edge to illuminate the ¯ow ®eld containing these smoke particles, a sharp contrast could be obtained between the duct walls and the smoke.
For each case in the experiment we ®rst impose a fully developed ¯ow in the entire test section and then turn on the DC power supplies to raise the bottom plate temperature. Meanwhile, the distilled water is
cir-culated over the top plate. Each case takes about 3 h to raise the Rayleigh number to the test point and another 2 h are needed to maintain the vortex ¯ow at the stable or statistical state. After this we start various measurements and ¯ow visualization. For the transient tests to observe the vortex ¯ow formation the Rey-nolds number is changed to the test points in about 10±20 s.
2.4. Analysis of data uncertainty
In order to reduce any possible bias and data re-duction errors between the true physical value and the readout of the sensors and transducers, the data acqui-sition systems and various instruments including multi-plexers (Computer Products RTP 743), a digital barometer (Drunk Products DPI-260), reference junc-tions (Celesco Transducer Products BRJ14), and Hast-ing mass ¯ow controllers (HFC-220E and HFC-220F) are calibrated and adjusted end to end on site by the Instrument Calibration Section, Q.A. Center, Chung Shan Institute of Science and Technology (CSICI), Taiwan, with the transfer standards that the cali-bration hierarchy can trace back to the standard of National Institute of Standard and Technology (NIST), USA. Before performing the end to end cali-bration all the sensors and transducers used were transported to CSIST for calibration or adjustment with the inter-lab standards based on the test point that will be encountered in the present test to get best calibration curve-®t data. The data reduction error is reduced further by using the best nonlinear least square calibration curve-®ts and by selecting a suitable gain code of the multiplexers.
Uncertainties in the Rayleigh number, Reynolds number and other independent parameters are calcu-lated according to the standard procedures established by Kline and McClintock [33]. The uncertainties of the thermophysical properties of the air are included in the analysis. The ¯uid properties are real time corrected based on the temperature and pressure detected at the inlet of the test section. In addition, the uncertainties of the control unsteadiness and temperature nonunifor-mity are accounted for in the evaluation of the data uncertainty. The analysis shows that the uncertainties of temperature, volume ¯ow rate, dimensions, Rey-nolds number and Rayleigh number measurements are estimated to be less than 20.058C, 21%, 20.05 mm, 22% and 25%, respectively.
3. Results and discussion
Various vortex ¯ow structures observed in the pre-sent study are summarized by a ¯ow regime map shown in Fig. 2. It is important to note from these
results that at this low Reynolds number, 3RReR5, the buoyancy driven vortex ¯ow is not completely dominated by the moving transverse rolls, as observed in the previous studies [17±20]. In fact, at slightly supercritical Rayleigh number mixed longitudinal/ transverse vortex ¯ow appears. This mixed vortex ¯ow may contain irregular cells in the downstream for Re 3 at Ra 2000: In addition, near and below the critical Rayleigh number we even have longitudinal vortex ¯ow in the duct. It is further noted that for cases near Ractransverse waves appear nonperiodically
in time in the duct core. Selected results from the pre-sent study will be examined in the following sections to illustrate the detailed characteristics of the above vor-tex ¯ow patterns.
3.1. Vortex ¯ow patterns near critical Rayleigh number To clearly illustrate the vortex ¯ow patterns induced at the low Rayleigh number around the critical level for the onset of convection Rac 1708), top view of
the ¯ow at the midheight of the duct y 0:5 for var-ious Re and Ra is shown in the following. For cases in which the ¯ow evolves to steady state the steady top view ¯ow photos are given. While for the other cases the ¯ow does not reach steady state at long time, the instantaneous photos at certain time instants in the statistical state are given. Firstly, the change in the vortex ¯ow patterns for Ra reduced from 4000 to 1200 at Re 5:0 is displayed in Fig. 3. The results indicate that for the highest Rayleigh number tested here, with Ra 4000, the vortex ¯ow in the entire duct is domi-nated by the downstream moving transverse rolls (Fig. 3(a)). The transverse rolls are generated period-ically in time in the entry region of the duct and move gradually downstream. The time period of this roll generation is 4.35 s. As the rolls leave the heated sec-tion, they gradually degenerate and ®nally disappear. It is noted that in the ranges of Re and Ra covered here the boundary between the pure transverse vortex ¯ow and the mixed longitudinal/transverse vortex ¯ow indicated in Fig. 2 can be roughly characterized by the
equation
Ra 2200 1:7Re4 1
At the lower buoyancy, with Ra 3000, the time peri-odic moving transverse rolls are weaker and only dom-inate in the duct core (Fig. 3(b)). Near each duct side two steady longitudinal vortex rolls prevail. Thus, we
have mixed longitudinal/transverse vortex ¯ow at Ra 3000: Note that the slow axial growth of the longitudi-nal rolls in the downstream direction squeezes the transverse rolls and causes them to bend slightly and to become shorter as they move downstream. More-over, at the juncture between the transverse and longi-tudinal rolls the vortex ¯ow is somewhat irregular. As the buoyancy is reduced further to Ra 2500, the
Fig. 3. Top view ¯ow photos at steady or statistical state for Re 5:0 and Ra (a) 4000, (b) 3000, (c) 2500, (d) 2000, (e) 1750, (f) 1650, (g) 1500 and (h) 1200.
transverse rolls weaken further and occupy a smaller region and more longitudinal rolls are induced near the duct sides (Fig. 3(c)). Note that for an even lower buoyancy with Ra 2000 the transverse rolls comple-tely disappear and steady longitudinal rolls dominate the entire vortex ¯ow at long time (Fig. 3(d)). It is of interest to observe that for a reduction of Ra to a slightly supercritical level of 1750 only two steady longitudinal rolls are induced near each duct side and in the duct core a number of downstream moving, weak transverse waves prevail (Fig. 3(e)). These traver-sing waves are generated nonperiodically in time in the duct entry. Note that the nonperiodic traversing trans-verse waves appear in a very narrow range of the Ray-leigh number around the curve ®tted from the present data in Fig. 2 as
Ra 1750 ÿ 540=Re2 2
As the buoyancy is lowered even further to the subcri-tical level with Ra 1650, 1500 and 1200 these waves disappear but a few longitudinal rolls are still induced in the side wall region (Fig. 3(f)±(h)). The appearance of the longitudinal rolls at the subcritical buoyancy is
further veri®ed by the end view ¯ow photos shown in Fig. 4. According to Fig. 3(d) the axial location at which each longitudinal roll begins to appear depends on its spanwise position. Based on the present data for all cases with steady longitudinal vortex ¯ow, this onset location can be correlated as
ln Raz 11:88 ÿ 0:0367 x20:471Re0:5z 3
Another new vortex ¯ow pattern containing longitudi-nal and transverse rolls and irregular cells is revealed in Fig. 5 by showing the top view ¯ow photos for var-ious Reynolds numbers at Ra 2000: These photos manifest that at Re 5:0 steady longitudinal rolls dominate in the ¯ow (Fig. 5(a)), as already discussed above. For the lower Re of 4.0 the buoyancy-to-inertia ratio is higher and the longitudinal rolls become unsteady (Fig. 5(b)). A further lowering of Re to 3.0 causes a few transverse rolls to appear in the entry portion of the duct (Fig. 5(c)). Besides, the longitudinal rolls in the duct core downstream of the transverse rolls become unstable and disintegrate into a number of irregular cells. Thus, at Re 3:0 the new vortex ¯ow is in the form of steady longitudinal rolls near the
duct sides, time periodic moving transverse rolls in the duct entry and irregular cells in the remaining region of the duct. Note that the irregular cells are prevalent in a larger region for a higher buoyancy-to-inertia ratio.
3.2. Temporal characteristics of vortex ¯ows
The temporal characteristics of the observed vortex
¯ows, revealed from the instantaneous temperature measurement, indicated that in the region dominated by the regular longitudinal rolls the ¯ow is steady, while the ¯ow is time periodic in the regular transverse vortex ¯ow. These features are illustrated in Figs. 6 and 7. The results in Fig. 6 for a pure transverse vor-tex ¯ow for Re 4 and Ra 4000 show that the entire ¯ow oscillates at the same frequency tp 5:6 s)
and amplitude in the duct core where the fully devel-oped transverse rolls prevail. In the duct entry where the transverse rolls are in developing stage and in the side wall region where the viscous eects are strong, weaker ¯ow oscillation is noted. It is of interest to notice that even in the region slightly upstream of the test section the ¯ow oscillates weakly at the same fre-quency as that in the test section. This is due to the presence of the returning ¯ow in that upstream unheated region associated with the main forced ¯ow blocked by the transverse rolls ahead of it. According to the present data, the oscillation frequency can be correlated as
F 0:45Re 6:70 10ÿ4Re3 4
Meanwhile, the convection speed of the transverse rolls Wrcan be estimated from the relation
Wr 1:3Wm 5
which is the same as that in our earlier study for a lower aspect ratio duct [20]. Additional thermocouple data given in Fig. 7 for a mixed vortex ¯ow with Re 5 and Ra 2500 further show that in the juncture between the longitudinal and transverse rolls the ¯ow is in an intensive irregular oscillation. Moreover, away from the vortex ¯ow in the duct entry the ¯ow is steady.
3.3. Formation of vortex ¯ow patterns
It is important in the fundamental study of heat transfer and ¯uid dynamics to unveil the processes through which the vortex structures discussed above are formed under the action of the buoyancy. In the study of the vortex ¯ow 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 inves-tigated so that the buoyancy-to-inertia ratio is rather low for a sucient period of time. Thus, the initial ¯ow t < 0 established in the duct is forced convection dominated. Then, at time t 0 the Reynolds number of the ¯ow is lowered quickly to the level for the speci®c case to be examined and maintained at this level thereafter for t > 0: Note that due to the ¯ow inertia, normally it takes about 10±20 s for the Rey-nolds number to be reduced to the required level.
Fig. 5. Top view ¯ow photos at steady or statistical state taken at y 0:5 for Ra 2000 and Re (a) 5.0, (b) 4.0 and (c) 3.0.
Firstly, the formation of the vortex ¯ow comprising of steady longitudinal rolls near the duct sides and nonperiodic traversing transverse waves in the duct core is illustrated in Fig. 8 by showing the top view
¯ow photos at selected time instants following the re-duction of Re from 20.0 to 4.0 for Ra 1750: The result in Fig. 8(a) indicates that the initial ¯ow at t 0 is nearly unidirectional with the presence of relatively
Fig. 6. Temporal structure of vortex ¯ow from (a) top view and (b) time records of air temperature at selected locations on the plane y 1=2 for Re 4 and Ra 4000 tp 5:6 s).
weak longitudinal rolls adjacent to the duct sides. Later at t 14 s new longitudinal rolls form adjacent to the existing ones. As time proceeds, additional longitudinal rolls are generated (Fig. 8(c)±(h)). The newly generated rolls are unstable in the beginning
(Fig. 8(c)±(f)). However, they gradually stabilize (Fig. 8(g)). It is of interest to note that in almost the entire transient stage nonperiodic traversing transverse waves appear in the region outside the longitudinal rolls, as evident from Fig. 8(b)±(h). The vortex ¯ow
Fig. 7. Temporal structure of vortex ¯ow from (a) top view and (b) time records of air temperature at selected locations on the plane y 1=2 for Re 5 and Ra 2500 tp 4:6 s).
pattern in the form of steady longitudinal rolls and traversing waves is thus formed. Note that due to the interaction between the rolls and waves, the longitudi-nal rolls at largest distance from the duct sides are un-stable to some degree (Fig. 8(e)±(h)). At the higher Ra of 2000 for Re 5:0 the traversing waves, however, disappear for t > 700 s and a stable longitudinal vortex ¯ow prevails in the duct. Moreover, for cases with RaR1650 at Re 4:0 and 5.0 and for RaR1500 at
Re 3:0 traversing transverse waves only appear during the early transient stage and stable longitudinal rolls exist at steady state in the duct.
Next, the formation of the time periodic moving transverse vortex ¯ow and the mixed longitudinal/ transverse vortex ¯ow is found to resemble that in our previous study for a lower aspect ratio duct A 12 [22].
Finally, the mixed longitudinal and transverse vortex
Fig. 8. Top view ¯ow photos showing the formation of nonperiodic traversing transverse waves and longitudinal rolls by lowering Re from 20.0 to 4.0 in 12 s for Ra 1750 at time t (a) 0 s, (b) 14 s, (c) 40 s, (d) 81 s, (e) 216 s, (f) 378 s, (g) 2706 s and (h) 4088 s.
¯ow containing irregular cells prevailed at lower buoy-ancy than that for the transverse vortex ¯ow is formed through a series of complicate processes shown in Fig. 9 for the case with Re reduced from 20.0 to 3.0 at Ra 2000: The results manifest that immediately after the Reynolds number is lowered, a transverse roll is generated at the duct inlet and meanwhile, longitudinal rolls are induced near the duct sides (Fig. 9(b)). How-ever, at this low buoyancy the transverse rolls repeat-edly generated at the inlet are weak and only exist in the duct core outside the longitudinal rolls (Fig. 9(c)).
Besides, these downstream moving transverse rolls are somewhat deformed at later time and become knotted (Fig. 9(c)±(e)). Moreover, the knots grow with time and in the adjacent transverse rolls they may contact with each other to form cellular vortex ¯ow in the duct core (Fig. 9(f) and (g)). At this stage the longi-tudinal rolls near the duct sides are also highly deformed despite of their continuing growth. In ad-dition, we note that the transverse rolls generated at the duct inlet become rather weak. In fact, they consist of a series of traversing transverse waves. These waves
Fig. 9. Top view ¯ow photos showing the formation of mixed longitudinal and transverse rolls and downstream irregular cells by lowering Re from 20 to 3 in 6 s at Ra 2000 and t (a) 0 s, (b) 11 s, (c) 14 s, (d) 21 s, (e) 30 s, (f) 44 s, (g) 64 s, (h) 81 s, (i) 106 s, (j) 187 s, (k) 276 s, (l) 795 s.
again become knotted and later degenerate into weak, cellular vortex ¯ow. As the weak transverse waves move downstream, the cellular vortex ¯ow ahead of it is pushed slowly out of the duct (Fig. 9(g) and (h)). For a further increase in time additional longitudinal rolls are gradually formed and eventually the down-stream portion of the duct is completely ®lled with longitudinal rolls (Fig. 9(i)±(l)). The irregular vortex cells and transverse waves only exist in the duct core upstream of the longitudinal rolls (Fig. 9(l)). Note that the period of time leading to the mixed vortex ¯ow containing longitudinal and transverse rolls and cells is substantially longer than that for the transverse vortex ¯ow.
4. Concluding remarks
Experimental ¯ow visualization combined with tran-sient temperature measurement have been conducted here to explore the buoyancy driven vortex ¯ow struc-tures in mixed convection of air through a bottom heated horizontal ¯at duct of a large aspect ratio A 16). Attention is paid to the ¯ow at a very low Reynolds number ranging from 3.0 to 5.0 and at the Rayleigh number around the critical level for the onset of convection ranging from 1200 to 4000, including the subcritical and supercritical states. Results from the present study reveal that both at subcritical and slightly supercritical Rayleigh numbers steady longi-tudinal rolls are induced in the duct even at such low Reynolds numbers. In addition to the moving trans-verse rolls and the mixed longitudinal/transtrans-verse rolls which are often observed in the low Reynolds number mixed convection in a horizontal ¯at duct, some new vortex ¯ow structures are noted in the present study. In particular, we identify two new structures, namely, the stable longitudinal rolls along with the nonperiodic traversing transverse waves and the mixed longitudi-nal/transverse rolls as well as irregular cells. A ¯ow regime map in terms of Ra vs. Re was given to delin-eate the induced vortex ¯ow patterns.
The formation processes leading to various vortex ¯ow patterns are rather complicate, including the gen-eration of the longitudinal and transverse rolls and tra-versing waves, splitting of rolls into cells and the reverse process of integrating cells into rolls, in ad-dition to moving and bending of transverse rolls. Acknowledgements
The ®nancial support of this study by the engineer-ing division of National Science Council of Taiwan, R.O.C. through the contract NSC83-0404-E009-054 is greatly appreciated.
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