W. S. Tseng
W. L. Lin
C. P. Yin
C. L. Lin
T. F. Lin
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.Stabilization of Buoyancy-Driven
Unstable Vortex Flow in Mixed
Convection of Air in a Rectangular
Duct by Tapering Its Top Plate
Stabilization of the buoyancy-driven unstable mixed convective vortex air flow in a bottom heated rectangular duct by tapering its top plate is investigated experimentally. Specifi-cally, the duct is tapered so that its aspect ratio at the duct inlet is 4 and gradually raised to 12 at the exit of the duct. In the study the secondary flow in the duct is visualized and the steady and transient thermal characteristics of the flow are examined by measuring the spanwise distributions of the time-average temperature. The effects of the Reynolds and Grashof numbers on the vortex flow structure are studied in detail. Moreover, the spanwise-averaged Nusselt numbers for the horizontal rectangular and tapering ducts are also measured and compared. Furthermore, the time records of the air temperature are obtained to further detect the temporal stability of the flow. Over the ranges of the Re and Gr investigated for 5⭐Re⭐102 and 1.0⫻104⭐Gr⭐1.7⫻105, the vortex flow induced in the rectangular duct exhibits temporal transition from a steady laminar to time periodic and then to chaotic state at increasing buoyancy-to-inertia ratio. Substantial change in the spatial structure of the vortex flow is also noted to accompany this temporal transi-tion. The results for the tapering duct indicate that more vortex rolls can be induced due to the increase in the aspect ratio of the duct with the axial distance. But the vortex rolls are weaker and are completely stabilized by the tapering of the top plate.
关S0022-1481共00兲70301-X兴
Keywords: Heat Transfer, Mixed Convection, Vortex
1 Introduction
At high buoyancy-to-inertia ratio frequently encountered in various heat transfer equipment, the buoyancy-driven secondary vortex flow in a forced laminar flow through a bottom heated rectangular duct is rather unstable. Heat transfer augmentation associated with the buoyancy-driven vortex flow is desirable and welcome in many technological applications in which the efficient energy transport is of major concern. However, in the chemical vapor deposition共CVD兲 processes used to grow thin crystal films on semiconductor substrates, the presence of the vortex flow at a high buoyancy and a low inertia for the Reynolds number ranging from 1 to 100 will result in a nonuniform chemical vapor deposi-tion, producing a thin crystal film of nonuniform thickness. More-over, the unsteady vortex flow will provoke a time-dependent deposition rate. These unsteady vortex flows are not welcome and should be avoided in the CVD processes. A simple mean often used to suppress and stabilize the vortex flow is to accelerate the main flow so that the buoyancy-to-inertia ratio decreases in the flow direction. This experimental study intends to explore the change in the vortex flow structure of air caused by the flow acceleration when the top plate of a bottom-heated horizontal rect-angular duct is slightly tapered.
In a horizontal parallel plane channel with the bottom plate at a higher uniform temperature than the top one by⌬T, the critical Rayleigh number for the onset of the vortex flow was found to be around 1708, as predicted from experimental measurements and linear stability theory共关1,2兴兲. Ostrach and Kamotani 关3兴 and Ka-motani and Ostrach关4兴 experimentally noted that the longitudinal vortex rolls appear when the temperature difference becomes
larger than a critical value corresponding to Ra⫽1708. Above the critical point, the heat transfer rate is increased by the thermal instability and the temperature field is strongly influenced by the motion of the vortex rolls. When Ra⬎8000, the vortex rolls be-come irregular. Hwang and Liu关5兴 visualized the onset of second-ary flow and showed that the wave number of vortex rolls remains constant along the flow direction. Experiments conducted by In-cropera et al.关6兴 and Maughan and Incropera 关7兴 disclosed four flow regimes along the bottom plate-laminar forced convection, laminar mixed convection, transitional mixed convection, and tur-bulent free convection. The transition to turtur-bulent flow was attrib-uted to the breakdown of the vortices due to the hydrodynamic instability. Besides, a correlation for the onset points was pro-posed. Criteria for the onset of vortex instability and the start of the transition from two-dimensional laminar flow to three-dimensional vortex flow in mixed convection air flow over an isothermally heated horizontal flat plate were established by Mo-harreri et al. 关8兴. The vortex flow regime started with a stable laminar flow region where vortices develop and grow gradually and ended with an unstable flow region where the vortices mix together and collapse to form a two-dimensional turbulent flow regime.
In the mixed convection of the nitrogen gas between two hori-zontal, differentially heated parallel plates, Chiu and Rosenberger 关9,10兴 showed that the critical Ra for the appearance of the trans-verse convection rolls increased with Re. They also found that at a high Ra, the longitudinal rolls were unsteady and snaking. Be-sides, they showed the regime of Ra and Re leading to a steady or unsteady flow. Nyce et al.关11兴 showed the variations of velocity field in both the axial and transverse directions which were con-sistent with the presence of both longitudinal roll and transverse wave instabilities. The transverse rolls were noted at a very low Reynolds number by Ouazzani et al.关12,13兴. They also refined the regime map to include the transverse rolls. The recent flow visualization from Yu et al.关14兴 showed that at a fixed Ra but in Contributed by the Heat Transfer Division for publication in the JOURNAL OF
HEATTRANSFER. Manuscript received by the Heat Transfer Division, Feb. 7, 1999; revision received, July 27, 1999. Associate Technical Editor: F. Cheung.
reducing Re the vortex flow transforms from the structure pre-vailed by the longitudinal rolls to transverse waves in sequence of stable longitudinal rolls, unstable longitudinal rolls, mixture of the longitudinal rolls and transverse waves, and transverse waves. Photographic results from Cheng and Shi关15兴 revealed the con-vective instability and chaotic phenomena caused by the buoyancy forces. Koizumi and Hosokawa关16兴 presented the sidewall tem-perature profile in a horizontal rectangular duct heated from below which simulated a horizontal thermal CVD reactor. They found that although the flow was unsteady but the time-averaged local mass transfer rate on the bottom wall can be almost uniform.
Recently, in a series of studies 共关17–20兴兲 we unraveled the complex flow characteristics in an air flow through a bottom-heated horizontal rectangular duct of aspect ratio AR⫽2 & 4. The formation, merging, and splitting of the longitudinal vortex rolls were numerically and experimentally predicted for the laminar steady, laminar time periodic, transitional quasi-periodic, and ir-regular chaotic flow at increasing buoyancy-to-inertia ratio.
The above literature review clearly reveals that there has been a substantial amount of research carried out in the past on various aspects of the vortex flow and heat transfer in mixed convection of gas in a horizontal rectangular duct. However, very little is known on the processes of how the vortex flow is suppressed by the flow acceleration resulting from reducing the cross-sectional area of the duct. An experimental investigation combining flow visualization and temperature measurements will be conducted in the present study to unravel the effects of tapering the top plate of rectangular duct on the detailed changes in the vortex flow struc-ture in a mixed convective air flow through a bottom heated trap-ezoid duct. Attention will be focused on the flow stabilization and weakening of the vortex flow by the acceleration of the primary flow for various Reynolds and Grashof numbers.
2 Experimental Apparatus and Procedures
2.1 Experimental Apparatus. Figure 1 shows a schematic arrangement of the mixed convective experimental apparatus es-tablished for the present study. The apparatus consists of three parts: wind tunnel, test section, and measuring probes along with a data acquisition unit. Two different test sections have been de-signed, one for the rectangular duct without tapering the top plate and another for the trapezoid duct with the top plate tapered downwards from the duct inlet to exit. The test section for the rectangular duct is constructed of 9-mm-thick plexiglass top and side walls to facilitate flow visualization. The rectangular cross section is 30 mm in height and 120 mm in width and the duct has a total length of 800 mm, providing an aspect ratio of AR⫽4 for the duct. The tapering duct is also 800 mm long and is made of
10-mm-thick plexiglass top and 9-mm-thick plexiglass side walls. The top plate is inclined downwards so that the aspect ratio of the duct cross section increases from 4 at the inlet to 12 at the exit with the duct height reduced from 30 mm at the inlet to 10 mm at the exit. The inclined angle of the top plate is rather small at 1.432 deg when measured from the horizontal. The details of the mixed convective loop were already described in Lin and Lin关19兴. Be-sides, the experimental procedures and method to analyze the tem-perature fluctuation were also given in that study.
2.2 Data Reduction for Heat Transfer Coefficient. The spanwise-averaged Nusselt number defined as
Nud⫽ hdin k ⫽ qconv⬙ din 共T¯w⫺Tin兲k (1) is obtained from measuring the spanwise-averaged bottom wall temperature T¯wand local convective heat flux qconv⬙ . Note that the heat transfer coefficient h is based on the temperature difference (T¯w⫺Tin) instead of (T¯w⫺Tb) since the heated section is not long (L/din⫽26.7) and the convective heat transfer in it mainly takes place in the entrance region. The surface energy balance relating the total energy dissipated in the heater plate due to the electric resistance heating per unit surface area qtot⬙ to the heat fluxes as-sociated with the convection from the surface to the flow qconv⬙ , net radiation heat loss from the surface qrad⬙ and conduction loss through the insulation qinsul⬙ is
qtot⬙⫽qconv⬙ ⫹qrad⬙ ⫹qinsul⬙ . (2) The convective heat flux is therefore determined by measuring the total dissipated heat flux and applying appropriate correlations for the nonconvective components. An implication of the expression in Eq.共2兲 is that although uniform heat generation qtot⬙ is achieved in the heated plate, variations in the radiation and conduction losses with the location induce nonuniformities in the convective heat flux. The axial variations in qconv⬙ were estimated to be less than 20 percent of the average qconv⬙ . The radiation and conduc-tion losses were estimated to range from 32 percent to 36 percent and 13 percent to 17 percent, respectively.
Uncertainties in the Nusselt, Reynolds, and Grashof numbers were estimated according to the standard procedures proposed by Kline and McClintock关21兴. This analysis indicated that through the combination of many measurements together with the uncer-tainties in the conductive and radiative losses, the unceruncer-tainties in Re, Gr, and Nudare 2 percent, 11 percent, and 4 percent, respec-tively.
3 Discussion of Experimental Results
In what follows, only a small sample of the results obtained are presented to illustrate the effects of the top plate tapering on the flow stabilization by examining the resulting vortex flow struc-tures and time history of the air temperature. In addition, the span-wise average Nusselt number affected by the plate tapering is also inspected. More detailed results are available from the thesis of Tseng关22兴.
3.1 Spanwise Average Heat Transfer Coefficient. The axial distributions of the steady spanwise average Nusselt number was obtained first for the limiting cases of forced convection with the Grashof number Gr (⫽gqconv⬙ din
4
/k2) below 2000 for the rectangular and tapering ducts. The data for the rectangular duct were very close to that calculated for the forced convection in that duct from the previous numerical computation 共关18兴兲. This was verified in our previous studies共关19,20兴兲. In the exit region the Nusselt number for the tapering duct is slightly above that for the rectangular duct due to the acceleration of the main flow.
Effects of the Grashof number on the time-averaged axial varia-tions of the spanwise average Nusselt number in the two different Fig. 1 Schematic diagram of the experimental apparatus
ducts are shown in Fig. 2 for Re⫽102. The results in Fig. 2共a兲 for the horizontal top plate indicate that at a given Gr near the duct entry the Nusselt number decreases monotonically in the flow direction. At a certain axial location Nudreaches a minimum and rises gradually. This location corresponds to the onset of the buoyancy-induced vortex flow. More specifically, the onset point is defined as the axial station where the spanwise average Nusselt number enhanced by the vortex flow exceeds that for the forced convection by three percent. At a higher Grashof number this onset location appears at a shorter distance from the duct inlet. The monotonic decay in the Nusselt number in the upstream is associated with the dominated laminar forced convection there. Beyond the onset point plumes of warm fluid rise from the heated surface resulting in a developing secondary flow and the plate temperature is reduced by the cool air descending from the duct core. Thus, the flow circulation provides an effective mechanism for heat transfer enhancement. As the secondary flow strengthens, the Nusselt number rises well above the forced convection limit. Further downstream the core flow has been warmed up and Nud reverses its trend to decrease in the flow direction. Thus, we have oscillatory Nuddistributions in the axial direction for a high Gr. The results for the tapering duct共Fig. 2共b兲兲 resemble those for the rectangular duct 共Fig. 2共a兲兲 except that in the tapering duct the onset of the vortex flow moves a little downstream as compared with that for the horizontal duct for the same Re and Gr. Note that the duct height at the duct inlet is used in defining Gr for the tapering duct. It is further noted that the axial rise in Nud due to the vortex flow are slightly smaller for the tapering duct, implying the induced secondary flow being weaker. Also note that Nudis
lower for the tapering duct in the downstream region of the duct. These results indicate that the Nusselt number increases with the Grashof number except in the region near the exit of the tapering duct. Near the exit region, the flow accelerates so rapidly that the transport process is dominated by forced convection. Therefore, the Grashof number only exhibits a mild effect on the heat trans-fer near the duct exit.
The data in Fig. 3 for illustrating the effects of the Reynolds number on the Nud distributions for both ducts indicate that the onset of secondary flow moves downstream at increasing Rey-nolds number for Gr fixed at 7.4⫻104. This trend is in compli-ance with the fact that the velocity boundary layer gets thinner at a higher Re and thereby increases its resistance to thermal insta-bility, and hence impedes secondary flow development. Oscilla-tory Nudvariations with the axial distance are also noted for these curves. More intense oscillations result for a higher buoyancy-to-inertia ratio. Moreover, at the same Re and Gr the onset point moves downstream in the tapering duct as compared with that for the corresponding case in the rectangular duct. Besides, Nud is also lower in the tapering duct.
It is of interest to note that the onset points for various cases covered here for both ducts can be correlated as
Grz⫽3637 Rez 1.66
(3) where Grzand Rezare the local Grashof and Reynolds numbers, Grz⫽gqconv⬙ z
4
/k2and Rez⫽w¯z/. Here w¯ is the local mean ve-locity of the air in the ducts, which increases linearly with the axial distance for the tapering duct. But for the rectangular duct w¯ is constant.
Fig. 2 Effects of Grashof number on the spanwise-averaged Nusselt number distribution for ReÄ102 for „a… rectangular duct and„b…tapering duct
Fig. 3 Effects of Reynolds number on the spanwise-averaged Nusselt number distribution for GrÄ7.4Ã104for„a…rectangular duct and„b…tapering duct
3.2 Vortex Flow Structure and Spanwise Temperature Distributions. Flow visualization was conducted to detect the resulting buoyancy-induced vortex flow patterns at steady or sta-tistical state for both rectangular and tapering ducts for various Reynolds and Grashof numbers. They were carried out by using a 1.5–3-mm plane light beam shone through the flow field contain-ing small incense smoke particles as light scattercontain-ing centers.
Side views of the instantaneous flow photos taken at the central vertical plane at x⫽6 cm for various Reynolds and Grashof num-bers clearly showed the uplift of the streamlines from the bottom plate at certain axial locations, suggesting the onset of the vortex flow there. The onset points were also found to advance upstream for a higher Grashof number or a lower Reynolds number. The photos are not shown here due to the limited availability of the space for the article. Tapering the top plate was noted to cause significant delay in the onset of vortex flow at low buoyancy-to-inertia ratio (Gr/Re2). The delay gradually diminished at increas-ing Gr/Re2. At a high Gr/Re2no noticeable delay in the onset of vortex flow was detected. Inspecting these side view flow photos for the low Reynolds and high Grashof numbers revealed the pres-ence of irregular recirculating cells and the associated secondary flow was somewhat unstable. This recirculating flow became highly irregular when reaching the downstream portion of the duct. It was also noted that at this high buoyancy-to-inertia ratio
the induced vortex flow was so strong that the forced flow enter-ing the duct was seriously blocked and hence was reversed near the duct inlet resulting in a returning flow there共关23兴兲. More spe-cifically, the returning flow was formed by a part of warm air ascending from the bottom plate slightly upstream of the onset point and penetrating towards the top plate. When the warm air reached the top plate, it split into two streams. One moved up-stream and another downup-stream along the top plate over a certain distance. Meanwhile the upstream moving stream was slowed down by the main forced flow. Finally, it merges with the main flow around the inlet of the heated section. Thus an elongated recirculating cell is formed in the duct entry. It is also noted that after the onset point of thermal instability, most of the test section is dominated by the natural convection for Gr/Re2⭓40.
To illustrate the effects of the top plate inclination on the de-tailed vortex flow structures, the instantaneous flow photos from the top and end views were also examined. It was noted from the results for the rectangular duct that at a low buoyancy-to-inertia ratio for Re⫽102 and Gr⫽2.5⫻104a symmetric pair of longitu-dinal rolls were induced near the side walls beginning at z ⬇15 cm. Slightly downstream at z⬇31 cm another symmetric pair of longitudinal rolls were generated. This four-roll structure prevailed beyond that. Raising Gr to 7.4⫻104 caused an earlier Fig. 4 „a…Instantaneous flow photos and„b…time-averaged spanwise
tempera-ture distributions atyÄ1.5 cm at selected cross sections in rectangular duct for ReÄ102 and GrÄ2.5Ã104
onset of the rolls. The vortex flow in the downstream still con-sisted of four-roll structure. Lowering the Reynolds number from 102 to 88 for Gr fixed at 7.4⫻104resulted in the generation of a new pair of longitudinal rolls so that there were six longitudinal rolls in the duct. But at this higher Gr/Re2 the vortex flow is slightly asymmetric with respect to the central vertical plane at x⫽6 cm. At an even higher Gr/Re2the longitudinal rolls were somewhat irregular and were obviously asymmetric. Additional flow photos for the very low Reynolds numbers of Re⫽15 and 9 indicated that the vortex flow still consisted of symmetric longi-tudinal rolls when Gr was not too high. Next, snapshots of the flow viewing from the top for the tapering duct at the same Re and Gr were inspected. Note that for the low buoyancy-to-inertial ratio case with Re⫽102 and Gr⫽2.5⫻104, a symmetric pair of longi-tudinal rolls induced in the entry half of the tapering duct were rather weak. At the higher Gr of 7.4⫻104a pair of stronger lon-gitudinal rolls were induced near the side walls beginning at z ⬇8 cm. At z⬇22 cm, two plumes of warm air rose near the cen-tral vertical plane (x⫽6 cm) and formed two new pairs of longi-tudinal rolls. Thus, beyond that we had three pairs of longilongi-tudinal rolls in the tapering duct. Lowering the Reynolds number from 102 to 88 for Gr fixed at 7.4⫻104again caused an earlier appear-ance of the three pairs of longitudinal rolls. At the even lower
Reynolds number of Re⫽51, three symmetric pairs of vortex rolls started to appear at z⬇15 cm. The results for the low Reynolds number of Re⫽15 and 9 with Gr fixed at 1.0⫻104for the tapering duct indicated that for Re⫽15 only one symmetric roll pair was induced near the side walls and no noticeable roll was induced in the center region of the heated section. The rolls near the side walls were suppressed by the accelerating air flow and were weaker than those in the rectangular duct at the same Re and Gr. At a lower Reynolds number of Re⫽9 two pairs of longitudinal rolls began to appear at z⬇16 cm, one adjacent to the center vertical plane at x⫽6 cm and another near the side walls. Note that for z⭓30 cm the two vortex pairs were significantly sup-pressed by the accelerating main forced flow and their strength got weaker with the downstream distance.
The above observation of the vortex structure changes with the Grashof and Reynolds numbers and with the duct tapering can be further illustrated by the end-view photos for the cross-plane sec-ondary flow at selected cross sections. The corresponding span-wise distributions of the time-averaged air temperature at the mid-height of the duct are also given to quantitatively show the intensity of the vortex rolls. Figure 4 distinctly manifests the evo-lution of the vortex pattern in the rectangular duct from the uni-Fig. 5 „a…Instantaneous flow photos and„b…time-averaged spanwise
tempera-ture distributions atyÄ1.5 cm at selected cross sections in tapering duct for ReÄ102 and GrÄ2.5Ã104
directional flow in the duct entry to the two-roll structure slightly downstream and then to the four-roll structure in the exit half of the duct for Re⫽102 and Gr⫽2.5⫻104. It is also noted from the flow photos and time-averaged temperature distributions that in the upstream (z⬍0.4 m兲 the rolls near the duct sides are stronger than those in the duct core. In the downstream the reverse is the case. The corresponding results for the tapering duct at the same Re and Gr are shown in Fig. 5. Note that in the tapering duct one pair of longitudinal rolls clearly exist near the side walls in nearly the entire test section. In the downstream the second pair of lon-gitudinal rolls are rather weak and can be barely seen in the pho-tos for z⫽0.5 and 0.7 m. It is also noted from the time-averaged temperature distributions共Figs. 4共b兲 and 5共b兲兲 that the strength of the vortex rolls for the tapering duct is much weaker than that in the rectangular duct at the same Re and Gr. Further comparison of the results for the higher buoyancies共Gr⫽4.2⫻104& 7.4⫻104兲 was also made. We noted that the photos for the rectangular duct 共Fig. 6共a兲兲 showed a similar vortex flow evolution to that at a lower buoyancy for Gr⫽2.5⫻104 except the appearance of an additional pair of small vortex rolls near the duct sides in the exit end of the duct. This newly formed vortex pair circulates down-wards near the side walls. It should be pointed out that this new vortex pair is small and mainly appears in the duct corners, and hence is not detected by the top view flow photos discussed above in which the secondary flow at the horizontal plane at the mid-height of the duct y⫽1.5 cm was pictured. Moreover, at these higher buoyancies the rolls in the core region grow significantly as the flow moves downstream. While in the tapering duct an in-crease of the Grashof number to 4.2⫻104resulted in three pairs of weak longitudinal rolls in the secondary half of the duct. The vortex rolls were obviously weaker than those in the horizontal
duct. For a higher Grashof number of Gr⫽7.4⫻104共Fig. 6共b兲兲 the vortex flow in the tapering duct consists of two weak longitudinal roll pairs in the entry region of the duct core (z⬍28 cm), three vortex roll pairs in the middle portion of the duct (28 cm⭐z ⭐55 cm兲, and four pairs in the exit end. Note that due to the flow acceleration in the axial direction associated with the top plate tapering the vortex intensity of the longitudinal rolls does not always increase with the axial distance. For example, the rolls at the cross section z⫽0.7 m are much weaker than those at the cross section z⫽0.6 m. These vortex rolls are also weaker than those in the horizontal duct at the same Re and Gr共Fig. 6共a兲兲.
The Reynolds number can have a dramatic impact on the vortex flow development, as discussed above. This is further manifested by inspecting the end-view photos for Gr⫽1.0⫻104and Re⫽15 and 9. Note that at these lower Reynolds numbers the vortex rolls in the rectangular duct get stronger with the axial distance but do not merge together共Fig. 7共a兲兲. Moreover, the degree of the vortex flow asymmetry is largest in the duct entry where the vortex flow is relatively weak. Downstream the four vortex rolls are in fact spanwisely symmetric. Note that in the tapering duct only one vortex pair forms at Re⫽15 共Fig. 7共b兲兲 and the vortex flow is much weaker. At Re⫽9 there were four rolls induced shortly after the flow entered the duct. But the rolls in the duct core were substantially suppressed by the top plate tapering to nearly disap-pear in the duct exit. The rolls near the duct sides were also weaker and are in triangular shape. Again the vortex intensity of the flow in the tapering duct is much weaker. For a further reduc-tion of the Reynolds number to 5 the vortex flow quickly evolved to one pair of strong, symmetric longitudinal rolls in the rectan-gular duct. In the tapering duct the four-pair structure prevailed in the entry half of the duct. Downstream the rolls in the duct core Fig. 6 Instantaneous and view flow photos at selected cross
sections for ReÄ102 and GrÄ7.4Ã104for„a…rectangular duct and„b…tapering duct
Fig. 7 Instantaneous end view flow photos at selected cross sections for ReÄ15 and GrÄ1.0Ã104 for„a… rectangular duct and„b…tapering duct
were significantly suppressed. These downstream rolls were slightly asymmetric. The vortex flow was weakened again by the top plate tapering but in a smaller degree at this low Reynolds number of Re⫽5.
To reveal the temporal characteristics of the vortex flow, the time variations of the detected instantaneous air temperature at selected locations for various cases for the rectangular and trap-ezoid ducts were examined. It was of interest to note that the flow oscillation in the rectangular duct at high buoyancy-to-inertia ra-tios was completely suppressed by the flow acceleration in the tapering duct. This clearly shows the effective stabilization of the buoyancy driven unstable flow by the top plate tapering.
4 Concluding Remarks
We have performed an experimental study to investigate the effects of the top plate tapering on the stabilization of the buoy-ancy driven vortex flow in mixed convection of air through the bottom heated horizontal rectangular and tapering ducts by sys-tematic measurement of the time-averaged spanwise temperature distributions and the time histories of the air temperature. Instan-taneous flow visualizations were also conducted. The experiments were carried out for the Reynolds number ranging from 5 to 102 and Grashof numbers from 1.0⫻104 to 1.7⫻105. The major re-sults can be brightly summarized in the following.
1 In the rectangular duct the onset of thermal instability was found to move upstream for increasing Grashof number and/or decreasing Reynolds number. The results for the tapering duct resemble those for the rectangular duct except that the onset point moves a little downstream as compared with that for the horizon-tal duct for the same Re and Gr.
2 Due to the increase of the aspect ratio from 4 at the duct inlet to 12 at the duct exit, more vortex rolls are induced in the tapering duct than in the rectangular duct in the downstream region.
3 In the tapering duct, the main forced flow accelerates so rap-idly in the downstream region that the vortex intensity of the longitudinal rolls are much weaker.
4 The unstable vortex flow at high buoyancy-to-inertia ratio in the rectangular duct can be completely stabilized by tapering the top plate over the range of the parameters covered in the present study.
During the course of this study it has been realized that al-though the top plate tapering results in the complete stabilization of the vortex flow, the weaker vortices still dominate the second-ary flow in the tapering duct. Methods to eliminate these vortices need to be investigated in the future.
Acknowledgment
The financial support of this study by the engineering division of National Science Council of Taiwan, R.O.C. through the con-tract NSC 87-2212-E-009-029 is greatly appreciated.
Nomenclature
AR ⫽ aspect ratio, b/d
b, d ⫽ duct width and duct height g ⫽ gravitational acceleration Gr ⫽ Grashof number, gqconv⬙ din4/k2 Grz ⫽ local Grashof number, gqconv⬙ z4/k2
h ⫽ local convection heat transfer coefficient k ⫽ thermal conductivity
L ⫽ length of the test section Nud ⫽ Nusselt number, hdin/k
Pr ⫽ Prandtl number,/␣ qconv⬙ ⫽ local convective heat flux
qinsul⬙ ⫽ local conduction heat loss through the insulation
qrad⬙ ⫽ local radiation heat loss from the surface
qtot⬙ ⫽ the total heat flux in the plate Ra ⫽ Rayleigh number, Gr•Pr Re ⫽ Reynolds number, windin/ Rez ⫽ local Reynolds number, w¯z/
T ⫽ temperature
w ⫽ velocity component in axial direction x, y, z ⫽ Cartesian coordinates
X, Y, Z ⫽ dimensionless Cartesian coordinate, x/din, y /din, and z/din
Z* ⫽ modified Z-coordinate, z/din•Re•Pr
 ⫽ thermal expansion coefficient ⫽ kinematic viscosity
Subscripts
b ⫽ of buck quantities
in ⫽ values at the inlet of the test section w ⫽ of heated wall quantities
Superscripts
⫺ ⫽ average value
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