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Thermal flows around a fully permeable short circular cylinder
International Journal of Heat and Mass Transfer 105 (2017) 196–206
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Thermal flows around a fully permeable short circular cylinder S. Schekman, T. Kim ⇑ School of Mechanical and Aeronautical Engineering, University of the Witwatersrand, Johannesburg, South Africa
a r t i c l e
i n f o
Article history: Received 18 August 2016 Received in revised form 28 September 2016 Accepted 28 September 2016
Keywords: Endwall heat transfer Horseshoe vortex Porous media Through-flow Wakes
a b s t r a c t Secondary flows and corresponding endwall heat transfer around a fully permeable (porous) short circular cylinder differ from those around an impermeable (solid) cylinder. There exists a through flow which is portion of the mainstream entering the front of the cylinder and exiting at the rear of the cylinder, and the alteration of an adverse pressure gradient along the plane of symmetry that causes three-dimensional boundary layer separation (or forms horseshoe vortices). This study experimentally demonstrates how these distinctive features vary horseshoe vortices, wake patterns, and endwall heat transfer around the porous short circular cylinder. Furthermore, the fluidic effect of the extent of the through-flow that is determined by the permeability of the porous cylinder on downstream heat transfer is discussed. Ó 2016 Published by Elsevier Ltd.
1. Introduction Fluid stream that approaches an object protruded from a surface experiences an adverse pressure gradient due to its eventual stagnation on it. At a certain location upstream from the object, the boundary layer separates from the surface. The separated boundary layer rolls up and forms a series of vortices, between the object and the point of separation. In accordance with Kelvin’s circulation theorem, the boundary layer vorticity cannot be destroyed. Instead, it is convected around each side of the object to form the two legs of the ‘‘horseshoe vortex” (see Fig. 1(a)). Due to the engineering significance of the horseshoe vortex, its numerous aspects have been studied [1–25]. Fig. 1(b) exemplifies the laminar horseshoe vortex that is composed of three counter rotating pairs. Each pair consists of a larger primary vortex, rotating in a clockwise direction – for the case of the flow moving from left to right – and a smaller secondary vortex that rotates in the opposite direction. It is necessary for the vortices to form in counter rotating pairs in order for the streamline topology, within the separated region, to be preserved [1]. The kinematics of these horseshoe vortices were explained by Baker [2] who detailed how each vortex illustrated in Fig. 1(b) moves and is fed by the freestream. The minimum number of vortices that can form is two (i.e., a single pair). A single vortex pair (V1 and V01 ) reported by Visbal [4] was observed at the Reynolds number of ReD = 500. As the Rey⇑ Corresponding author. E-mail address: [email protected] (T. Kim). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.09.089 0017-9310/Ó 2016 Published by Elsevier Ltd.
nolds number increases to ReD = 1500, two vortex pairs are formed until three vortex pairs are formed for the higher Reynolds number of ReD = 2500. The existence of more than 3 vortex pairs is unlikely due to the limited space between the point of boundary layer separation and the cylinder leading edge. Baker [2] showed that the number of vortex pairs depends on not only the Reynolds number but also the boundary layer displacement thickness for a short cylinder as summarized in Fig. 2. As the Reynolds number is increased, the primary vortex (V1) becomes unstable, followed by its break-up into two vortices; each with a more stable vortex core [5]. If the Reynolds number is fixed, an increased displacement thickness acts to move the boundary layer separation point further upstream [6]. It is possible that increasing the distance between the separation point and cylinder leading edge, would in some way either delay the instability in the primary vortex or allow for a vortex to remain stable at a higher Reynolds number at its core [6]; resulting in fewer vortex pairs forming. It has been shown that the horseshoe vortices will not form at all if a cylinder span-to-diameter ratio (or an aspect ratio, l/D) is too small. Akkoka [7] and Mendez et al. [8] both conducted numerical and experimental studies for aspect ratios of 0.116 6 l/ D 6 0.265 and Reynolds numbers of 1200 6 ReD 6 1460. They showed that under these conditions, no horseshoe vortices would appear at the juncture. For any l/D < 0.28, the boundary layers of the upper and lower endwalls would intersect, likely preventing the formation of the horseshoe vortices. For l/D > 1.0, the effect of the cylinder aspect ratio ceases to have a noticeable effect on the separation point or any of the other flow features [9].
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Nomenclature A Cp D h H l NuD ReD S SP T U u Uc V
attachment point pressure coefficient cylinder diameter, m convective heat transfer coefficient, W/(m2 K) channel height, m cylinder length (or span), m Nusselt number Reynolds number separation point saddle point endwall surface temperature, K mainstream velocity, m/s local axial velocity, m/s centerline velocity at y = H/2, m/s vortex
W x y z
channel width, m a longitudinal axis coinciding with fluid stream direction an axis coinciding with the channel height, protruded cylinder span, and normal to the endwall surface a transverse axis coinciding with the channel width
Greek symbols s wall shear stress, N/m2 q air density, kg/m3 ⁄ d boundary layer displacement thickness, m xs position of boundary layer separation, m
Fig. 2. Variation in number of vortices with a change in Reynolds number and boundary layer displacement thickness where l/D = 0.5 [2] and l is the span of the cylinder.
Fig. 1. Horseshoe vortices: (a) visualized horseshoe vortices [1], (b) three vortex pair streamline pattern on the plane of symmetry and endwall for a laminar horseshoe vortex system where V denotes the vortex, S denotes the separation point, A indicates the attachment point, and SP is the saddle point (adapted from Baker [2,3]).
For higher Reynolds numbers, the boundary layer displacement thickness would be smaller and the minimum aspect ratio required for horseshoe vortices to form would, therefore, be smaller as well. Akkoka [7] showed that for aspect ratios of l/D P 0.35, the horse-
shoe vortex system around the cylinder juncture becomes stable. This was taken as an indication that the vortex system was fully formed with no interaction occurring between the horseshoe vortices forming at the upper and lower endwalls. For this reason, most studies on the horseshoe vortices and corresponding heat transfer effects, make use of obstacles with l/D > 0.35, with many using l/D > 0.5 [2,10]. The topology and kinematics discussed so far refer to stable horseshoe vortices whereby the vortices do not move or oscillate. Once the Reynolds number increases above a certain value, typically ReD 3000, the vortices begin to oscillate [2]. A splitting and shedding of the vortices downstream of the cylinder is also reported to occur during these oscillations. There are conflicting reasons given for why the vortices begin to oscillate. It has been concluded, however, by Baker [2], Thomas [11], Agui and Andreopoulos [12], and Fu and Rockwell [13] that the unsteady process is produced by a natural instability of the horseshoe vortex system and not by Karman-vortex shedding in the wake of the cylinder. Higher Reynolds numbers of the approaching flow result in the approaching boundary layer becoming turbulent, the transition occurring for 400 < Red < 1000 with the flow being accepted as fully turbulent for any Reynolds number Red > 1000 [9]. When the system becomes turbulent, the flow structure becomes irregular. Time averaged flow visualization on the plane of symmetry has been stated to either indicate no vorticity occurring [14] or form only a single vortex pair [18,19]. Praisner and Smith [17] visualized
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two vortex pairs with a time-averaged measurement. Baker [3] only observed the separation lines S and S1 (in Fig. 1(b)) on the endwall surface for the turbulent horseshoe vortex system. The absence of the separation line S2, which is necessary for a 3 vortex pair system, was understood to indicate that only a two-vortex pair system was present. Praisner and Smith [17] also performed instantaneous measurements of the streamlines in the plane of symmetry. Only the primary vortex, V1 and secondary vortex, V01 were identified. Additional vortices, V2 and V3, are possibly present as well at certain points in time. No satisfactory explanation is given as to the reason why the vortex system suddenly shifts to having only two-vortex pairs. It is then possible, and more likely that threevortex pairs still exist but vortices V2 and V3 become more time dependent; being absorbed into one another and ejected, dissipating in the flow regime. These behaviors of vortices are observable even in the laminar regime at high Reynolds numbers, typically just before the flow becomes turbulent [18,21,22]. The vortices V2 and V3 are then present in a turbulent horseshoe vortex system but not all the time; multiple instantaneous flow visualizations being required to ascertain their positions. As it is already a very weak vortex, V3 would then likely disappear when taking time-averaged measurements, explaining its complete absence in previous publications. Hung et al. [20] showed, however, that the relation between flow and surface topologies means that the information of the flow field can only be extrapolated from the endwall flow visualization with extreme care. No stable vortex system with a turbulent boundary layer has ever been observed [9]. The presence of horseshoe vortices between the separation point and cylinder leading edge results in a large amount of entrainment near the endwall. This is due to the mainstream being entrained in each vortex, as seen in Fig. 1. This downstream movement prevents the entrained fluid stream from becoming trapped in the separated region. The result of this would be an expected increase in local heat transfer upstream of the cylinder; heat being transferred to the fresh fluid stream as it passes the endwall before being dissipated downstream of the cylinder (if the heat is input through the endwall). Measurements conducted by Praisner et al. [20,24] and Chyu et al. [23] showed an increase in heat transfer between the boundary layer separation point S, and the cylinder leading edge. A large decrease near the cylinder leading edge is as a result of the fluid stream being trapped in the corner vortex, V00 [22]. Unlike the primary vortices, the corner vortex entrains very limited fresh stream from the mainstream. While the upstream flow pattern for a horseshoe vortex system is well-documented, the downstream flow pattern is scarcely investigated. The main effects of horseshoe vortices, namely an increase in drag and shear stresses on a base plate around the object, are largely dependent on the upstream flow around it. The oscillation of the vortices is also due to the upstream flow rather than instabilities in the cylinder wake. Most of the studies that have reported on the downstream flow patterns typically performed flow visualization near the endwall. An example of the flow field at y/D = 0.05 (immediately above an endwall or a base plate) for ReD = 6150 is shown in Fig. 3. At this Reynolds number, the approaching boundary layer was turbulent [21]. The predominant feature of the downstream flow field is the vortex denoted as VD. The position of this vortex is largely dependent on the Reynolds number. At ReD = 1500, corresponding to a single vortex pair system [4], the vortex was positioned at x/ D = 1.5 from the cylinder axis. At ReD = 4000, for which threevortex pairs existed, the vortex was located approximately at x/ D = 1.2 downstream. At ReD = 6150, corresponding to a turbulent horseshoe vortex system, the vortex was observed at x/D = 0.9.
Fig. 3. Streamline patterns in cylinder near wakes at y/D = 0.05 (above an endwall), based on Laser-Doppler Velocimetry by Sahin [21] at ReD = 6150.
These values are interpreted from the figures reported by Sahin [21]. The horseshoe vortices that form upstream of the cylinder move downstream and become entrained in the vortex VD. This stimulates circulatory motion in the cylinder wake resulting in the formation of the saddle points SP3, SP4 and SP5, and two distinct flow regions, Wake Region I and II (as labeled in Fig. 3). Wake Region I forms between the cylinder circumference and SP4, and experiences very little fresh stream from the mainstream. Conversely, Wake Region II, which forms between SP4 and SP5, experiences high levels of flow entrainment. Wake Region I (see Fig. 3) is subjected to a limited amount of fluid being entrained from the mainstream, resulting in a decreased heat transfer. Wake Region II (see Fig. 3) receives a larger amount of fluid entrainment due to VD and subsequently a higher local heat transfer is present. Based on the large effect that horseshoe vortices have on the local flow characteristics around an object being discussed hitherto, there is a great interest in manipulating the formation of the horseshoe vortices. Besides the manipulation of the flow conditions such as Reynolds number and boundary layer thickness, other methods investigated focus on the object itself. Typically, these investigations looked into the effect that the object shape had on the horseshoe vortices by Chyu and Natarajan [23], Prainser et al. [22], Khan et al. [15] and Simpson [18]. While a circular cylinder is the most popular object’s shape, the other shapes include cubes, diamonds, pyramids, hemispheres, and airfoils. The different profiles were shown to affect the adverse pressure gradients, heat transfer around the objects and downstream wake profiles; blunt objects such as cubes tended to result in lower heat transfer and longer wake profiles than the circular cylinder. Another potential method for manipulating the formation of the horseshoe vortices is through the use of a porous medium. This research has in the past focused on unrelated topics such as seepage from streams bounded by porous banks [24] and displacement of oil from sandstones [25]. More related research was carried out by Khashehchi et al. [26] who made use of a partially permeable cylinder and focused only on the wakes at Reynolds numbers up to ReD = 10000. It was observed that the porous layer extended the wake regions behind the cylinder. However, no further effects on (endwall) heat transfer and fluidic mechanism inside the porous layer were investigated. The present study squarely addresses these issues experimentally: how the mainstream leaked through the fully porous cylinder alters the wakes and consequently how these wakes affect the endwall heat transfer around the porous cylinder. To this end, a series of endwall flow visualization, endwall heat transfer and pressure and wake measurements around three selected porous cylinders having different pore densities (10 pores per inch (PPI), 20 PPI and 30 PPI) but with a fixed porosity of 0.94, has been conducted at a turbulent Reynolds number of ReD = 1.0 104. Prior to experiments, the permeability of each porous cylinder that determines the amount of the mainstream leaked through the fully permeable cylinder was calculated based on pressure drop data.
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2. Experimental details 2.1. Test rig and cylinder samples An open suction-type wind-tunnel was used for static pressure/ velocity measurements, and endwall surface flow/temperature mappings. Fluid stream that is drawn by a centrifugal fan passes through a contraction, honeycomb and parallel flow section. At the beginning of the parallel flow section, the mainstream develops. A schematic of the test section with a cylinder sample is illustrated in Fig. 4. Dimensions of a wide rectangular test section are W H = 0.4 m (width) 0.045 m (height). Four test samples were tested (Fig. 5): one impermeable cylinder made from a pure copper (reference) and three copper porous cylinders cut using an electric discharge machine. The fully porous cylinders have the same porosity (e = 0.94) but the pore size of each cylinder (or pore density) varies to be 10 pores per inch (PPI), 20 PPI and 30 PPI. It should be noted that the ligaments of the present porous cylinders are hollow associated with their fabrication process. All the test cylinders have the same dimensions such as the diameter of D = 42 mm, the length (of span) of l = 45 mm. Therefore, the span-to-diameter (i.e., the aspect ratio) is 1.07, classified as a short cylinder. A blockage ratio (diameter-to-rectangular flow channel width, D/W) is 0.105.
2.2. Endwall surface flow visualization using an oil-dye mixture technique Endwall surface flow pattern was visualized using an oil-dye mixture technique in the wind-tunnel. Florescent powder was mixed with a light diesel fuel, and the mixture was then applied to the cylinder surface. When the aerodynamic shear stresses redistribute the mixture, visualization of the surface flow patterns is realized. Before the application of the mixture, the surface is
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painted in black to enhance the reflectivity when using ultraviolet light to illuminate the surface flow pattern while it is being photographed. More details associated with this flow visualization technique can be found in Ref. [27]. 2.3. Pneumatic pressure and velocity measurements Static pressure variation along the plane of symmetry was measured by pressure tappings. Steel tubes with an inner diameter of 0.6 mm were inserted in the drilled holes along the plane of symmetry. Each tube was then connected to a differential pressure transducer (DSATM, Scanivalve Inc.). Axial velocity component of the mainstream was traversed to quantify boundary layer (displacement) thickness at x/D = ()4.5 using a total pressure probe with a flattened tip which allows pressure at 0.23 mm away from the endwall to be measured (the closest data point). The origin of the cylinder sample coincides with xaxis (i.e., x = 0). A Pitot tube is positioned at x/D = 2.5 upstream of the cylinder axis and at the mid-span and mid-width of the flow channel to monitor the mainstream’s centerline velocity. The Reynolds number based on the cylinder diameter D and centerline velocity was fixed at ReD = 1.0 104 (a sub-critical flow regime). The pneumatic pressure tappings and stagnation pressure probe are connected to the differential pressure transducer which sends data via a TCP/IP protocol to a data acquisition personal computer. 2.4. Endwall surface temperature mapping using an infrared thermography A constant heat flux is generated by a film heating element (A = 0.3 m 0.3 m) that is attached to the outer surface of the substrate. A film heat flux sensor is installed to measure a portion of the generated heat flux by the film heating element that leaves through the outer surface of the substrate i.e., q00 loss. The net heat flux (q00 ) used to calculate heat transfer coefficient (h) is estimated as:
Fig. 4. Photograph and schematic of the test rig showing a suction-type wind-tunnel with a contraction and a test section accommodating a short single cylinder.
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Fig. 5. Tested circular cylinders: (a) Impermeable (solid) cylinder, (b) 30 PPI porous cylinder (e = 0.94), (c) 20 PPI porous cylinder (e = 0.94), (d) 10 PPI porous cylinder (e = 0.94).
q00net ¼
Q in q00loss A
ð1Þ
where Qin the total input power calculated from the measured input current and voltage to the film heater and A is the heating area. The input power (Qin) was kept constant during the entire study e.g., 18 W (or q00 in = 200 W/m2). The heat loss (q00 loss) at this set Reynolds number was measured to be approximately 5.0% of the total input heat flux. Local temperature distribution was mapped using an infrared (IR) thermal camera (FLIR T640). A polymer based upper substrate is used as a viewing window for the IR camera. The infrared (IR) window used in the present study was made from a plastic (Edmund OpticsTM), had 60% of transmittance for the 8–14 lm wavelength and its thickness was 0.45 mm. For the current setup, a drop in radiation intensity across the window was observed. To account for the intensity attenuation across the window, a series of calibrations were carried out. A systematic intensity attenuation that results in the temperature drop of 6.0 K is obtained in the present Reynolds number. The surface temperature data only along the plane of symmetry was extracted to calculate local Nusselt numbers as:
Nuðx; 0; 0Þ ¼
hðx; 0; 0Þ kf =D
ð2Þ
where kf is the thermal conductivity of air evaluated at a film temperature ((Tin + Tm)/2) and the convective heat transfer coefficient along the plane of symmetry, h(x,0,0) was calculated as:
hðx; 0; 0Þ ¼
q00net ðT s ðx; 0; 0Þ T in Þ
the value used, an uncertainty in kf was calculated to be ±3%. The uncertainty in the cylinder diameter (D) was ±0.05%. The uncertainty in the Nusselt number was, therefore, calculated to be within ±4.0%. 2.5. Data reduction parameters and measurement uncertainty The Reynolds number ReD is defined based on the cylinder diameter D and the centerline axial velocity Uc measured at the mid-height of the flow channel with no pressure gradient as:
ReD ¼
qU c D l
ð4Þ
where l and q are the viscosity and density of air, respectively. Local static pressure along the plane of symmetry is evaluated using the wall pressure coefficient defined as:
Cp ¼
pðx=DÞ pref
qU 2c =2
ð5Þ
where p(x/D) and pref are the measured static pressure and reference pressure (at x/D = ()4.5) and Uc is the centerline velocity measured at x/D = ()4.5. Uncertainty associated with the Reynolds number and the wall pressure coefficient are estimated following a method reported by Coleman and Steele [29] (based on 20:1 odds) and found to be within 1.8% and 1.8%, respectively. Note that only random errors are considered here. It is assumed that the bias (systematic) errors could be minimized by careful calibrations.
ð3Þ
where Tin is the temperature of the mainstream measured at x/ D = 2.5 upstream of the cylinder axis by a bead T-type thermocouple and Tm is an area averaged surface temperature from the IR image. The experimental uncertainty of the Nusselt number was estimated using a method reported by Holman and Gajda [28] based on 20:1 odds. For heat flux measurement, the error associated with the voltmeter scanner was estimated to be less than ±5 lV, resulting in an error less than ±6 W/m2 for heat flux. The resolution of the temperature readings from the temperature scanner was found to be ±0.1 K for each thermocouple. This same resolution was applied to the infrared camera as the thermocouples were used in its calibration. The uncertainty in the heat transfer coefficient (h) was calculated from these values to be ±0.42 W/(m2 K). The thermal conductivity of air (kf) was averaged along the temperature range. Based on the difference between the actual value and
3. Thermal flows around an impermeable short cylinder (reference case) A reference case for thermal flows around a short circular cylinder was established for a Reynolds number of ReD = 1.0 104. The endwall flow visualization and heat transfer measurements were conducted. 3.1. Endwall flow pattern The surface flow pattern around a short cylinder, which is impermeable, is shown in Fig. 6, depicting a well-known horseshoe vortex system clearly marked by oil-dye mixtures. It is expected that horseshoe vortex system would be in the turbulent regime at the chosen Reynolds number. Since only the time averaged behavior of the horseshoe vortex system is captured, the flow pattern (horseshoe vortex) is symmetric with respect to the x-axis.
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Fig. 7. Temperature distribution around an impermeable short cylinder at ReD = 1.0 104.
Time averaging also makes the identification of flow features such as attachment points A0, A1 and A2 as well as the separation points S1 and S2, as seen in Fig. 1(b), difficult. The overall endwall surface flow pattern in Fig. 6 consistently shows topological features previously reported [2,23]. A separation point appears to be located at about x/D = 1.2 upstream from the cylinder axis (labeled as ‘‘S”). This point lies close to the position predicted by Baker [3] that the boundary layer separates from the endwall at x/D = ()1.12. In this photograph, the separation lines are indicated by brighter lines as a result of accumulated dye particles. The two legs of the horseshoe vortex merge at approximately x/ D = 1.75 downstream, forming a saddle point (SP5). Another saddle point (SP4) is observed immediately behind the cylinder approximately at x/D = (+)0.75. It may appear as a nodal point as indicated by the accumulated oil-dye particles, configuring an isolated island along the cylinder axis. It was observed, however, during the test that two distinct islands of oil-dye formed apart with respect to the x-axis but once the fluid stream was stopped these two islands melded together. The accumulation of the oil-dye corresponds to the high region of flow entrainment in Wake Region II, shown in Fig. 3. The above features observed in the present reference case (in Fig. 6) are topologically consistent with those illustrated in Fig. 3.
4.1. The alteration of basic fluidic features The alteration of fluidic features around a porous cylinder due to the mainstream that partially leaks through the cylinder, forming a ‘‘through-flow” can be indicated by some upstream flow properties such as an upstream velocity profile (or boundary layer thickness) and adverse pressure gradient. Fig. 8(a) shows velocity profiles along the y-axis (channel height) at x/D = ()4.5 on the plane of symmetry where velocities are normalized by the centerline velocity Uc (i.e., Uy/H = 0.5) for the empty channel case. Data from only a lower half of the channel
Impermeable cylinder Permeable cylinder, 20 PPI Empty channel x
0.6 x
0.5 0.4
y/H
Fig. 6. Endwall flow pattern and temperature distribution for an impermeable copper short cylinder at ReD = 1.0 104.
of 30 PPI, 20 PPI and 10 PPI. Furthermore a comparison of the basic fluidic features upstream of the impermeable and porous cylinders were made.
0.3 0.2 0.1 0x 0
0.2
0.4
3.2. Endwall heat transfer
Comparisons to the reference case were made through the use of endwall flow visualization and heat transfer data for fully permeable cylinders with the fixed porosity of 0.94 and pore densities
0.8
1
(a) 1
Leading edge
Impermeable cylinder Permeable cylinder, 20 PPI 0.8 Adverse pressure Skin friction gradient 0.6 0.4 0.2 0 -6
4. Thermal flows around porous short circular cylinders
x x
Cp
The temperature distribution on the endwall captured by the pre-calibrated infrared thermal camera is shown Fig. 7. Black strips that appeared in the IR image are the steel rods that support the IR window film. Since local vortex flow features act to enhance local heat transfer, there is a direct link whose comparison can be made. The local temperature in the regions influenced by the horseshoe vortex such as that downstream of the separation point, S, and around the cylinder’s vertex is substantially lower than the rest region. Near the saddle point, SP4, and Wake Region II, the local temperature is higher than the other regions, indicating a lower heat transfer region.
x
0.6 u/Uc
x x x x x x x x x x x x x x x x x x x x x x xx xx xx x xx xx xx xx x xx xx
-5
-4
-3 x/D
-2
-1
0
(b) Fig. 8. Approaching flow properties: (a) velocity profiles at x/D = ()4.5 measured from the cylinder axis, (b) stream-wise pressure gradients.
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height are displayed since the velocity profile was confirmed to be symmetric with respect to the mid-span of the cylinder in the x–z plane. The centerline velocity Uc in the plane of symmetry decreases as the flow is decelerated by the presence of the cylinder and finally stagnates on the cylinder. However, the overall velocity profile appears to remain unchanged. To detail the changes, the displacement thickness of the developed boundary layer (d⁄) was calculated based on the data in Fig. 8 (a) as:
d ¼
Z 0
H=2
uðyÞ dy 1 Uc
ð6Þ
The results are listed in Table 1. In comparison with the solid (impermeable) cylinder, the porous cylinder gives rise to a slightly thinner displacement thickness, which is, however, thicker than that of the empty channel. The pressure distribution upstream of the cylinder until the cylinder’s leading edge in the plane of symmetry is plotted in Fig. 8(b). Before x/D = ()4.5, the static pressure is decreased although its extent is marginal. This decrease in the static pressure is due to pressure loss resulting from skin friction. Farther downstream from which, the Cp value begins to monotonically and significantly increase up to the nearest measurement point at x/D = ()0.75 (from the axis of the cylinder). This pressure rise (or adverse pressure gradient) is mainly due to the stagnation of the mainstream on the cylinder. With the porous cylinder having the pore density of 20 PPI, a similar pressure variation is observed along the plane of symmetry. The only difference is a reduced adverse pressure gradient from the upstream location at x/D = ()2.0 until the leading edge of the cylinder (x/D = ()0.5). 4.2. Endwall thermal flow patterns The slight changes in the boundary layer profile and adverse pressure gradient discussed with Fig. 8 may influence horseshoe vortices to be formed around the porous cylinders. Furthermore, a portion of the mainstream enters the porous cylinder forming a through-flow, and is expected to cause the alteration of the horseshoe vortices and wakes. Fig. 9 exhibits the endwall flow pattern visualized for the three fully permeable (porous) cylinders: 30 PPI (Fig. 9(a)), 20 PPI (Fig. 9 (b)), and 10 PPI (Fig. 9(c)). A common noticeable endwall flow pattern is the shortening of the distance between the separation point (S) and the leading edge of the cylinder. For the impermeable cylinder, the point at which the boundary layer separated from the endwall (S) was identified at x/D = ()1.2. For the 30 PPI porous cylinder, the separation point was identified at x/D = ()1.0. The allowance for the through-flow by the porous cylinder would decrease the adverse pressure gradient upstream of the cylinder, thereby causing the boundary layer separation to be delayed. For the 20 PPI and 10 PPI porous cylinders, the separation points were identified at x/D = ()0.9 and x/D = ()0.75, respectively. A lower pore density results in a larger decrease in the adverse gradient, and subsequent further delay in the separation of the boundary layer. Furthermore, the low shear stress region immediately downstream of the cylinder is elongated. A further elongation occurs as the pore density is decreased (i.e., the pore size becomes bigger).
Table 1 Boundary layer displacement thickness calculated at x/D = ()4.5. Cylinder sample
d⁄ [mm]
D/d⁄
Impermeable (solid) cylinder Permeable cylinder, 20 PPI Empty channel
1.81 1.78 1.67
23.2 23.6 25.2
Fig. 9. Endwall flow pattern for permeable short cylinders at ReD = 1.0 104: (a) for 30 PPI, (b) for 20 PPI, (c) for 10 PPI.
The horseshoe vortices enhance endwall heat transfer especially at the fore cylinder part as seen by a low temperature region there in Fig. 7. The overall endwall temperature distribution is similar to the endwall flow patterns in Fig. 6 for the impermeable cylinder. With the porous cylinder of 30 PPI (Fig. 10(a)), the variation of local temperature at the fore region of the porous cylinder is not distinguishable although that at the aft cylinder is marked by a high temperature region. However, with the porous cylinder of 20 PPI (Fig. 10(b)) and 10 PPI (Fig. 10(c)), the thinning of the low temperature region upstream of the cylinder becomes obvious. The elongated high temperature regions behind the cylinder coincide with the low shear stress region in Fig. 9. 4.3. Wakes behind the porous cylinders Before identifying the thermo-fluidic effects the through-flow might have on endwall heat transfer, further fluidic evidence of the through-flow needs to be understood.
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x/D = 3.0
x/D = 5.0
z/D
2.5 2 1.5 1 0.5 0 -0.5
0 u/Uc 1
0
1
(a) x/D = 3.0
x/D = 5.0
z/D
2.5 2 1.5 1 0.5 0 -0.5
0 u/Uc 1
0
1
(b) x/D = 3.0
x/D = 5.0
z/D
2.5 2 1.5 1 0.5 0 -0.5
0 u/Uc 1
0
1
(c) x/D = 3.0
x/D = 5.0
z/D
2.5 2 1.5 1 0.5 0 -0.5
0 u/Uc 1
0
1
(d) Fig. 11. Wake velocity profiles measured at x/D = (+)3.0 and x/D = (+)5.0 (downstream) from the cylinder axis: (a) solid cylinder, (b) porous cylinder with 30 PPI, (c) porous cylinder with 20 PPI, (d) porous cylinder with 10 PPI.
Fig. 10. Temperature distribution around permeable short cylinders: (a) for 30 PPI, (b) for 20 PPI, (c) for 10 PPI.
The wake profile measured at the downstream position, x/ D = 5.0, shows that both the impermeable cylinder and the porous cylinder with 30 PPI have a similar profile (Fig. 11(a, b)). Slightly upstream at x/D = (+)3.0 (closer to the aft cylinder), the profile for the impermeable cylinder is still approximately the same; the downstream wake having mostly re-established following the disturbance by the cylinder. For the 30 PPI porous cylinder, however, there is a marked change at x/D = (+)3.0, the velocity at the cylinder axis being approximately half of that at x/D = (+)5.0; the effects of the porous cylinder on the downstream wake last longer. Based on the wake profile data in Fig. 11(a, b), fluidic sketches are presented in Fig. 12(a, b). A significantly different profile is measured behind the porous cylinder with 20 PPI and 10 PPI at both downstream positions. At x/D = (+)3.0, erratic negative pressure readings were recorded close to the cylinder axis signifying that the fluid stream was moving upstream – the reverse flow. This would be because of the presence of vortices in the cylinder wake, as noted by Sahin [21]; these vor-
tices having moved further downstream for the lower pore density porous cylinder. At x/D = (+)5.0, there is no reverse flow observed; the axial velocity magnitude being lower for the 10 PPI porous cylinder than for the 20 PPI porous cylinder. The lower density porous cylinder results in the wake taking longer to re-establish as illustrated in Fig. 12(c, d), which is consistent with the observation made by Khashehchi et al. [26]. In summary, the wake profiles at the mid-span of the cylinder for the impermeable and porous cylinders are observed to be similar to those on the endwall: the through-flow within the porous cylinder resulting in a delay of the wake re-establishment with a lower pore density resulting in a longer delay. 4.4. Flow resistance in the porous cylinder It has been observed that the mainstream that partially leaks through the porous cylinder forming a through-flow causes: (a) the thinning of locally enhanced heat transfer region in front of the cylinder’s leading edge, and (b) an enlarged ‘‘thermally” less active region behind the cylinder and whose longitudinal coverage is elongated as more flow leaks through the porous cylinder.
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The permeability was then calculated from the modified Darcy equation [30] as,
dp 1 1 C F qu ¼ þ pffiffiffiffi dx lu K K l
ð7Þ
where dp/dx is the pressure drop per unit length along the nominal flow direction, CF is the inertial coefficient, u is the mean flow velocity. By plotting the quantity (dp/dx)(1/lu) as a function of qu/l for the range of flow rates considered, the two parameters are calculated graphically as shown in Fig. 13 where permeability is the intercept and the inertial coefficient is the slope. Results are summarized in Table 2. Higher permeability that implicitly indicates lower flow resistance is obtainable from a porous sample with bigger pore diameter for a given porosity. 4.5. Through-flow vs. wakes
Fig. 12. Sketch of flow patterns around the cylinder; (a) solid cylinder, (b) porous cylinder with 30 PPI, (c) porous cylinder with 20 PPI, (d) porous cylinder with 10 PPI.
The amount of the through-flow within the porous cylinder varies with the pore density. Within the porous cylinder, the flow resistance exists and is proportional to the thickness of the porous cylinder. Therefore, the highest resistance is encountered along the plane of symmetry. As a result, the mainstream that enters tends to detour to exit the porous cylinder via a flow path with less flow resistance. By the time when the through-flow exits the porous cylinder, the momentum is substantially dissipated. As the pore density becomes lower, the more through-flow can penetrate the porous cylinder due to a less flow resistance within the porous cylinder. The through-flow within the porous cylinder then becomes more straight or parallel to the x-axis. Even for a porous medium at a fixed porosity, pressure drop (or flow resistance) within the porous medium varies with the pore density. For the present three pore density samples, permeability (K) that is an imaginary length scale but an inversely and implicitly indicative of flow resistance was calculated from the pressure drop data as a function of mean (or nominal) flow velocity. In the parallel flow configuration, the pressure drop of the three porous samples was measured prior to the present study. A higher pressure drop is obtained from a porous sample with a smaller pore size for a fixed porosity (e = 0.94). Topologically, a porous sample with a smaller pore size has thinner ligaments.
The flow momentum that is introduced into the cylinder wake after a substantial dissipation within the porous cylinder at a certain exit angle, delays the horseshoe vortices from folding into form the downstream flow pattern as seen in Fig. 3. This delay moves the point at which the mainstream merges downstream (SP5) and moves the vortex VD and saddle point SP4 downstream (the horseshoe vortices being absorbed into VD). The downstream movement of these vortices means that Wake Region I is lengthened. This region having very little fresh fluid entrainment between the cylinder surface and SP4 (referring to Fig. 3). This region in Fig. 9(a–c) was identified by the lack of any significant disturbance to the oil-dye by the endwall shear stresses. The lower the pore density, the lower the resistance of that is encountered by the through-flow, the higher the momentum of the through-flow exiting the cylinder and subsequently the further downstream the saddle points SP4 and SP5. For the 30 PPI porous cylinder, Fig. 9(a), the size of the pooled island of oil-dye downstream of the cylinder is of a relatively similar size as that for the impermeable cylinder as seen in Fig. 6. As the pore density is increased, a notable increase in the size of the pooled oil-dye is observed. It is unlikely that this is due to a strengthening of Wake Region II with a decrease in the pore density but rather because of the increased distance between Wake Region II and the cylinder surface. The larger the distance, the more oil-dye that is carried towards the region of high fluid entrainment. As stated previously, it was observed during the tests that two distinct islands of oil-dye formed in Wake Region II but once the fluid stream stopped, these two islands melded together. Fig. 9(c)
Fig. 13. Graphical deduction of permeability (K) and inertial coefficient (CF) from the measured pressure drop (Dp/L) and mean velocity (u) for three copper porous samples (e = 0.94).
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Table 2 Properties of the copper porous samples where ks = 398 W/(m K) where the surface area density (or surface-to-volume ratio) asfof the porous samples was obtained according to the method used by Fuller et al. [31] based on the cubic model since the ligaments are hollow. Porosity, e Pores per inch (PPI) Pore diameter, dp[mm] Surface area density, asf[m2/m3] Permeability, K [m2]107 K1/2 [m] 104 Inertial coefficient, CF
0.94 (measured) 10 20 2.54 1.27 2286 5370 3.63 1.45 6.02 3.81 0.115 0.0848
30 0.85 8009 0.712 2.67 0.0952
still shows this to a small degree on the downstream side of the pooled dye.
Fig. 15. Schematic of the downstream flow topology on the endwall and trend in Nusselt number.
4.6. Wakes vs. endwall heat transfer To quantify how the through-flow alters local surface temperature distribution (or endwall heat transfer), the temperature data was extracted from Figs. 7 and 10 along the plane of symmetry. The results presented in the Nusselt number are plotted in Fig. 14 and show that local heat transfer is decreased as approaching the leading edge (LE) of the cylinder but turns to increase for all four included cases. Their turning point appears to be associated with the separation point (S). From the impermeable cylinder to porous cylinders (30 PPI ? 20 PPI ? 10 PPI), the turning point moves slightly towards the LE. After each turning point, local heat transfer increases steeply, having the highest Nusselt number obtained from the impermeable cylinder whereas the lowest heat transfer is obtainable from the porous cylinder with 10 PPI. A dramatic difference can be seen on local heat transfer downstream of the cylinder. Substantially low local heat transfer take place behind the cylinder that is porous. The trend of the Nusselt number for the impermeable cylinder (Fig. 14) exhibits a minimum point at a position of x/D = (+)0.75. Comparing this to the endwall flow pattern (Fig. 6), this position approximately coincides with the identified position of SP4. A similar minimum point is seen for all three porous cylinder cases (30 PPI, 20 PPI and 10 PPI). However, their positions are upstream of saddle point SP4, within Wake Region I (Fig. 3). After the minimum point the Nusselt number, for the impermeable cylinder, increases before reaching a maximum at x/D = (+)1.5. Comparing to the endwall flow pattern in Fig. 6, this maximum point occurs at the identified position of the saddle point SP5. The Nusselt numbers for the porous cylinders also exhibit an upward trend after the minimum point, the data does not
100
Impermeable Cu cylinder Porous Cu cylinder, 30 PPI Porous Cu cylinder, 20 PPI Porous Cu cylinder, 10 PPI
90
NuD
80 70 60 50 40 -1.5
-1
-0.5
0
0.5
x/D
1
1.5
2
2.5
Fig. 14. Nusselt numbers along the x-axis in the plane of symmetry for the solid cylinder (as reference) and Cu porous with three pore densities.
reach a maximum value but is assumed to follow a similar trend as that for the impermeable cylinder; the saddle points SP5 for the porous cylinders are all outside the region that temperature measurements were made. A schematic of the flow topology on the endwall and trend in the Nusselt number for the porous cylinders is shown in Fig. 15. In an area of high fluid entrainment from the core flow, such as Wake Region II (Fig. 3), there will be a higher degree of heat transfer at the endwall. Wake Region II (as per Fig. 3) and is noted as having high levels of fluid entrainment. Greater heat transfer would be expected at SP5 as there is both upstream flow, towards Wake Region II, and downstream flow, following the mainstream. This is why there is a local maximum seen in Fig. 15 for the impermeable cylinder, at SP5. In an area of low fluid entrainment (Wake Region I), the fluid stream is trapped meaning that there will be a decrease in the heat transfer. The minimum heat transfer would then occur somewhere this region. For the impermeable cylinder, the point of minimum heat transfer occurs at the downstream side of Wake Region I, close to SP4 (not coinciding with SP4). The minimum heat transfer for the 30 PPI cylinder occurs at approximately x/D = 0.6 (Fig. 14), placing it closer to the cylinder circumference than SP4, further upstream than that of the permeable cylinder. As the pore density is decreased, the point of minimum heat transfer moves downstream. For the 10 PPI porous cylinder, the minimum is closer to SP4 than the cylinder circumference. 5. Conclusions A series of experiments to investigate the alteration of horseshoe vortices formed around a short circular cylinder due to the mainstream that is partially leaked through the cylinder, have been performed at a turbulent Reynolds number. The findings in this study are summarized as follows. (1) The presence of the mainstream that is partly leaked through the porous cylinder causes a delay of the separation point of the incoming boundary layer flow at the fore cylinder and the elongation of the wakes at the aft cylinder. An adverse pressure gradient along the plane of symmetry is reduced as well. (2) The amount of through-flow is determined by the permeability (or pressure drop): a lower pore density porous cylinder for a fixed porosity allows more through-flow, leading to the more greatly elongated wakes. The flow that loses its substantial momentum within the cylinder meets the external mainstream at a certain exit angle when leaving the cylinder (blowing effect), causes the elongated wakes.
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(3) In the elongated wake regions, substantially lower endwall heat transfer takes place, posing thermally less active regions since the lower shear stresses exist there. (4) Before the inner saddle point (SP4) on the endwall, a local minimum of heat transfer exists whereas a local maximum endwall heat transfer coincides with the outer saddle point (SP5) along the plane of symmetry. (5) Regarding the endwall heat transfer modification, the presence of the through-flow within the cylinder (or an object) acts negatively by (a) narrowing a thermally active region under the influence of horseshoe vortices at the fore cylinder and (b) elongating thermally less active regions at the aft cylinder.
References [1] H. Werle, On the flow of fluids made visible, ONERA (1975). [2] C.J. Baker, The laminar horseshoe vortex, J. Fluid Mech. 95 (1978) 347–367. [3] C.J. Baker, The turbulent horseshoe vortex, J. Wind Eng. Ind. Aerodyn. 6 (1979) 9–23. [4] M.R. Visbal, Structure juncture flows, AIAA 29 (1991) 1273–1282. [5] C.J. Baker, The oscillation of horseshoe vortex systems, J. Fluids Eng. 113 (1991) 489–495. [6] C.J. Baker, Vortex Flow Around the Bases of Obstacles, St Catherine’s College, Cambridge, 1978. [7] A. Akkoka, Computational Modeling of Turbulent Heat Transfer in Plate Fin and Tube Heat Exchangers, Cukuova University, Institute of Natural and Applied Sciences, Department of Mechanical Engineering, Adana, Turkey, 2004. [8] R.R. Mendez, M. Sen, K.T. Yang, R. Mcclain, Effect of fin spacing on convection in a plate fin and tube heat exchanger, Int. J. Heat Mass Transfer 43 (2000) 39– 51. [9] F. Ballio, C. Bettoni, S. Franzetti, A survey of time-averaged characteristics of laminar and turbulent horseshoe vortices, J. Fluids Eng. 120 (1998) 233–242. [10] K. Marakkos, J.T. Turner, Vortex Generation in the Cross Flow Around a Cylinder Attached to an End Wall, School of Mechanical, Aerospace and Civil Engineering, University of Manchester, Manchester, UK, 2005. [11] A. Thomas, The unsteady characteristics of laminar juncture flow, Phys. Fluids 30 (1987) 283–285.
[12] J.H. Agui, J. Andreopoulos, Experimental investigation of a three-dimensional boundary layer flow in the vicinity of an upright wall mounted cylinder, J. Fluids Eng. 114 (1992) 566–576. [13] H. Fu, D. Rockwell, Shallow flow past a cylinder: transition phenomena at low Reynolds number, J. Fluid Mech. 540 (2005) 75–97. [14] W.A. Eckerle, J.K. Awad, Effect of freestream velocity on the three-dimensional separated flow region in front of a cylinder, J. Fluids Eng. 113 (1991) 37–44. [15] M.J. Khan, A. Ahmed, J.R. Trosper, Dynamics of the juncture vortex, AIAA 33 (7) (1995) 1273–1278. [16] R. Abid, R. Schmitt, Experimental study of a turbulent horseshoe vortex using a three component laser velocimeter, AIAA 86 (1986). [17] T.J. Praisner, C.R. Smith, The dynamics of the horseshoe vortex and associated endwall heat transfer – part II: time-mean results, J. Turbomach. 128 (2006) 755–762. [18] R.L. Simpson, Junction flows, Annu. Rev. Fluid Mech. 33 (2001) 415–443. [19] M.J. Khan, A. Ahmed, Topological model of flow regimes in the plane of symmetry of a surface mounted obstacle, Phys. Fluids 17 (2005). [20] C.M. Hung, C.H. Sung, C.L. Chen, Computation of saddle point of attachment, AIAA 30 (1992) 1561–1713. [21] B. Sahin, N.A. Ozturk, C. Gurlek, Horseshoe vortex studies in the passage of a model plate-fin-and-tube heat exchanger, Int. J. Heat Fluid Flow 06 (005) (2007) 2007. [22] T.J. Praisner, C.V. Seal, L. Takmaz, C.R. Smith, Spatial-temporal turbulent flowfield and heat transfer behavior in end-wall junctions, Int. J. Heat Fluid Flow 18 (1997) 142–151. [23] M.K. Chyu, V. Natarajan, Heat transfer on the base surface of threedimensional protruding elements, Int. J. Heat Mass Transfer 39 (14) (1996) 2925–2935. [24] J. Bear, D. Zaslavsky, S. Irmay, Physical Principles of Water Percolation and Seepage, UNESCO, Paris, 1968. [25] J.E. Bobek, P.T. Bail, Model studies of oil displacement from thin sands by vertical water influx adjacent shales, J. Petrol. Technol. (1961) 950. [26] M. Khashehchi, I.A. Abdi, T. Roesgen, K. Hooman, A comparison between the wake behind finned and poroused circular cylinders in cross-flow, Exp. Therm. Fluid Sci. 52 (2014) 328–338. [27] T. Kim, T.J. Lu, S.J. Song, Application of Thermo-Fluidic Measurement Techniques: An Introduction, Butterworth-Heinemann, 2016. [28] J.P. Holman, Gajda Jr., Experimental Methods for Engineers, eighth ed., McGraw-Hill, New York, 2012. [29] H.W. Coleman, W.G. Steele, Experimentation Validation, and Uncertainty Analysis for Engineers, third ed., John Wiley & Sons, Hoboken, NJ, 2009. [30] D.A. Nield, A. Bejan, Convection in Porous Media, fourth ed., Springer, 2013. [31] A.J. Fuller, T. Kim, H.P. Hodson, T.J. Lu, Measurements and interpretation of the heat transfer coefficients of metal foams, IMechE Part C Mech. Eng. Sci. 219 (2) (2005) 183–191.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Thermal flows around a fully permeable short circular cylinder S. Schekman, T. Kim ⇑ School of Mechanical and Aeronautical Engineering, University of the Witwatersrand, Johannesburg, South Africa
a r t i c l e
i n f o
Article history: Received 18 August 2016 Received in revised form 28 September 2016 Accepted 28 September 2016
Keywords: Endwall heat transfer Horseshoe vortex Porous media Through-flow Wakes
a b s t r a c t Secondary flows and corresponding endwall heat transfer around a fully permeable (porous) short circular cylinder differ from those around an impermeable (solid) cylinder. There exists a through flow which is portion of the mainstream entering the front of the cylinder and exiting at the rear of the cylinder, and the alteration of an adverse pressure gradient along the plane of symmetry that causes three-dimensional boundary layer separation (or forms horseshoe vortices). This study experimentally demonstrates how these distinctive features vary horseshoe vortices, wake patterns, and endwall heat transfer around the porous short circular cylinder. Furthermore, the fluidic effect of the extent of the through-flow that is determined by the permeability of the porous cylinder on downstream heat transfer is discussed. Ó 2016 Published by Elsevier Ltd.
1. Introduction Fluid stream that approaches an object protruded from a surface experiences an adverse pressure gradient due to its eventual stagnation on it. At a certain location upstream from the object, the boundary layer separates from the surface. The separated boundary layer rolls up and forms a series of vortices, between the object and the point of separation. In accordance with Kelvin’s circulation theorem, the boundary layer vorticity cannot be destroyed. Instead, it is convected around each side of the object to form the two legs of the ‘‘horseshoe vortex” (see Fig. 1(a)). Due to the engineering significance of the horseshoe vortex, its numerous aspects have been studied [1–25]. Fig. 1(b) exemplifies the laminar horseshoe vortex that is composed of three counter rotating pairs. Each pair consists of a larger primary vortex, rotating in a clockwise direction – for the case of the flow moving from left to right – and a smaller secondary vortex that rotates in the opposite direction. It is necessary for the vortices to form in counter rotating pairs in order for the streamline topology, within the separated region, to be preserved [1]. The kinematics of these horseshoe vortices were explained by Baker [2] who detailed how each vortex illustrated in Fig. 1(b) moves and is fed by the freestream. The minimum number of vortices that can form is two (i.e., a single pair). A single vortex pair (V1 and V01 ) reported by Visbal [4] was observed at the Reynolds number of ReD = 500. As the Rey⇑ Corresponding author. E-mail address: [email protected] (T. Kim). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.09.089 0017-9310/Ó 2016 Published by Elsevier Ltd.
nolds number increases to ReD = 1500, two vortex pairs are formed until three vortex pairs are formed for the higher Reynolds number of ReD = 2500. The existence of more than 3 vortex pairs is unlikely due to the limited space between the point of boundary layer separation and the cylinder leading edge. Baker [2] showed that the number of vortex pairs depends on not only the Reynolds number but also the boundary layer displacement thickness for a short cylinder as summarized in Fig. 2. As the Reynolds number is increased, the primary vortex (V1) becomes unstable, followed by its break-up into two vortices; each with a more stable vortex core [5]. If the Reynolds number is fixed, an increased displacement thickness acts to move the boundary layer separation point further upstream [6]. It is possible that increasing the distance between the separation point and cylinder leading edge, would in some way either delay the instability in the primary vortex or allow for a vortex to remain stable at a higher Reynolds number at its core [6]; resulting in fewer vortex pairs forming. It has been shown that the horseshoe vortices will not form at all if a cylinder span-to-diameter ratio (or an aspect ratio, l/D) is too small. Akkoka [7] and Mendez et al. [8] both conducted numerical and experimental studies for aspect ratios of 0.116 6 l/ D 6 0.265 and Reynolds numbers of 1200 6 ReD 6 1460. They showed that under these conditions, no horseshoe vortices would appear at the juncture. For any l/D < 0.28, the boundary layers of the upper and lower endwalls would intersect, likely preventing the formation of the horseshoe vortices. For l/D > 1.0, the effect of the cylinder aspect ratio ceases to have a noticeable effect on the separation point or any of the other flow features [9].
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Nomenclature A Cp D h H l NuD ReD S SP T U u Uc V
attachment point pressure coefficient cylinder diameter, m convective heat transfer coefficient, W/(m2 K) channel height, m cylinder length (or span), m Nusselt number Reynolds number separation point saddle point endwall surface temperature, K mainstream velocity, m/s local axial velocity, m/s centerline velocity at y = H/2, m/s vortex
W x y z
channel width, m a longitudinal axis coinciding with fluid stream direction an axis coinciding with the channel height, protruded cylinder span, and normal to the endwall surface a transverse axis coinciding with the channel width
Greek symbols s wall shear stress, N/m2 q air density, kg/m3 ⁄ d boundary layer displacement thickness, m xs position of boundary layer separation, m
Fig. 2. Variation in number of vortices with a change in Reynolds number and boundary layer displacement thickness where l/D = 0.5 [2] and l is the span of the cylinder.
Fig. 1. Horseshoe vortices: (a) visualized horseshoe vortices [1], (b) three vortex pair streamline pattern on the plane of symmetry and endwall for a laminar horseshoe vortex system where V denotes the vortex, S denotes the separation point, A indicates the attachment point, and SP is the saddle point (adapted from Baker [2,3]).
For higher Reynolds numbers, the boundary layer displacement thickness would be smaller and the minimum aspect ratio required for horseshoe vortices to form would, therefore, be smaller as well. Akkoka [7] showed that for aspect ratios of l/D P 0.35, the horse-
shoe vortex system around the cylinder juncture becomes stable. This was taken as an indication that the vortex system was fully formed with no interaction occurring between the horseshoe vortices forming at the upper and lower endwalls. For this reason, most studies on the horseshoe vortices and corresponding heat transfer effects, make use of obstacles with l/D > 0.35, with many using l/D > 0.5 [2,10]. The topology and kinematics discussed so far refer to stable horseshoe vortices whereby the vortices do not move or oscillate. Once the Reynolds number increases above a certain value, typically ReD 3000, the vortices begin to oscillate [2]. A splitting and shedding of the vortices downstream of the cylinder is also reported to occur during these oscillations. There are conflicting reasons given for why the vortices begin to oscillate. It has been concluded, however, by Baker [2], Thomas [11], Agui and Andreopoulos [12], and Fu and Rockwell [13] that the unsteady process is produced by a natural instability of the horseshoe vortex system and not by Karman-vortex shedding in the wake of the cylinder. Higher Reynolds numbers of the approaching flow result in the approaching boundary layer becoming turbulent, the transition occurring for 400 < Red < 1000 with the flow being accepted as fully turbulent for any Reynolds number Red > 1000 [9]. When the system becomes turbulent, the flow structure becomes irregular. Time averaged flow visualization on the plane of symmetry has been stated to either indicate no vorticity occurring [14] or form only a single vortex pair [18,19]. Praisner and Smith [17] visualized
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two vortex pairs with a time-averaged measurement. Baker [3] only observed the separation lines S and S1 (in Fig. 1(b)) on the endwall surface for the turbulent horseshoe vortex system. The absence of the separation line S2, which is necessary for a 3 vortex pair system, was understood to indicate that only a two-vortex pair system was present. Praisner and Smith [17] also performed instantaneous measurements of the streamlines in the plane of symmetry. Only the primary vortex, V1 and secondary vortex, V01 were identified. Additional vortices, V2 and V3, are possibly present as well at certain points in time. No satisfactory explanation is given as to the reason why the vortex system suddenly shifts to having only two-vortex pairs. It is then possible, and more likely that threevortex pairs still exist but vortices V2 and V3 become more time dependent; being absorbed into one another and ejected, dissipating in the flow regime. These behaviors of vortices are observable even in the laminar regime at high Reynolds numbers, typically just before the flow becomes turbulent [18,21,22]. The vortices V2 and V3 are then present in a turbulent horseshoe vortex system but not all the time; multiple instantaneous flow visualizations being required to ascertain their positions. As it is already a very weak vortex, V3 would then likely disappear when taking time-averaged measurements, explaining its complete absence in previous publications. Hung et al. [20] showed, however, that the relation between flow and surface topologies means that the information of the flow field can only be extrapolated from the endwall flow visualization with extreme care. No stable vortex system with a turbulent boundary layer has ever been observed [9]. The presence of horseshoe vortices between the separation point and cylinder leading edge results in a large amount of entrainment near the endwall. This is due to the mainstream being entrained in each vortex, as seen in Fig. 1. This downstream movement prevents the entrained fluid stream from becoming trapped in the separated region. The result of this would be an expected increase in local heat transfer upstream of the cylinder; heat being transferred to the fresh fluid stream as it passes the endwall before being dissipated downstream of the cylinder (if the heat is input through the endwall). Measurements conducted by Praisner et al. [20,24] and Chyu et al. [23] showed an increase in heat transfer between the boundary layer separation point S, and the cylinder leading edge. A large decrease near the cylinder leading edge is as a result of the fluid stream being trapped in the corner vortex, V00 [22]. Unlike the primary vortices, the corner vortex entrains very limited fresh stream from the mainstream. While the upstream flow pattern for a horseshoe vortex system is well-documented, the downstream flow pattern is scarcely investigated. The main effects of horseshoe vortices, namely an increase in drag and shear stresses on a base plate around the object, are largely dependent on the upstream flow around it. The oscillation of the vortices is also due to the upstream flow rather than instabilities in the cylinder wake. Most of the studies that have reported on the downstream flow patterns typically performed flow visualization near the endwall. An example of the flow field at y/D = 0.05 (immediately above an endwall or a base plate) for ReD = 6150 is shown in Fig. 3. At this Reynolds number, the approaching boundary layer was turbulent [21]. The predominant feature of the downstream flow field is the vortex denoted as VD. The position of this vortex is largely dependent on the Reynolds number. At ReD = 1500, corresponding to a single vortex pair system [4], the vortex was positioned at x/ D = 1.5 from the cylinder axis. At ReD = 4000, for which threevortex pairs existed, the vortex was located approximately at x/ D = 1.2 downstream. At ReD = 6150, corresponding to a turbulent horseshoe vortex system, the vortex was observed at x/D = 0.9.
Fig. 3. Streamline patterns in cylinder near wakes at y/D = 0.05 (above an endwall), based on Laser-Doppler Velocimetry by Sahin [21] at ReD = 6150.
These values are interpreted from the figures reported by Sahin [21]. The horseshoe vortices that form upstream of the cylinder move downstream and become entrained in the vortex VD. This stimulates circulatory motion in the cylinder wake resulting in the formation of the saddle points SP3, SP4 and SP5, and two distinct flow regions, Wake Region I and II (as labeled in Fig. 3). Wake Region I forms between the cylinder circumference and SP4, and experiences very little fresh stream from the mainstream. Conversely, Wake Region II, which forms between SP4 and SP5, experiences high levels of flow entrainment. Wake Region I (see Fig. 3) is subjected to a limited amount of fluid being entrained from the mainstream, resulting in a decreased heat transfer. Wake Region II (see Fig. 3) receives a larger amount of fluid entrainment due to VD and subsequently a higher local heat transfer is present. Based on the large effect that horseshoe vortices have on the local flow characteristics around an object being discussed hitherto, there is a great interest in manipulating the formation of the horseshoe vortices. Besides the manipulation of the flow conditions such as Reynolds number and boundary layer thickness, other methods investigated focus on the object itself. Typically, these investigations looked into the effect that the object shape had on the horseshoe vortices by Chyu and Natarajan [23], Prainser et al. [22], Khan et al. [15] and Simpson [18]. While a circular cylinder is the most popular object’s shape, the other shapes include cubes, diamonds, pyramids, hemispheres, and airfoils. The different profiles were shown to affect the adverse pressure gradients, heat transfer around the objects and downstream wake profiles; blunt objects such as cubes tended to result in lower heat transfer and longer wake profiles than the circular cylinder. Another potential method for manipulating the formation of the horseshoe vortices is through the use of a porous medium. This research has in the past focused on unrelated topics such as seepage from streams bounded by porous banks [24] and displacement of oil from sandstones [25]. More related research was carried out by Khashehchi et al. [26] who made use of a partially permeable cylinder and focused only on the wakes at Reynolds numbers up to ReD = 10000. It was observed that the porous layer extended the wake regions behind the cylinder. However, no further effects on (endwall) heat transfer and fluidic mechanism inside the porous layer were investigated. The present study squarely addresses these issues experimentally: how the mainstream leaked through the fully porous cylinder alters the wakes and consequently how these wakes affect the endwall heat transfer around the porous cylinder. To this end, a series of endwall flow visualization, endwall heat transfer and pressure and wake measurements around three selected porous cylinders having different pore densities (10 pores per inch (PPI), 20 PPI and 30 PPI) but with a fixed porosity of 0.94, has been conducted at a turbulent Reynolds number of ReD = 1.0 104. Prior to experiments, the permeability of each porous cylinder that determines the amount of the mainstream leaked through the fully permeable cylinder was calculated based on pressure drop data.
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2. Experimental details 2.1. Test rig and cylinder samples An open suction-type wind-tunnel was used for static pressure/ velocity measurements, and endwall surface flow/temperature mappings. Fluid stream that is drawn by a centrifugal fan passes through a contraction, honeycomb and parallel flow section. At the beginning of the parallel flow section, the mainstream develops. A schematic of the test section with a cylinder sample is illustrated in Fig. 4. Dimensions of a wide rectangular test section are W H = 0.4 m (width) 0.045 m (height). Four test samples were tested (Fig. 5): one impermeable cylinder made from a pure copper (reference) and three copper porous cylinders cut using an electric discharge machine. The fully porous cylinders have the same porosity (e = 0.94) but the pore size of each cylinder (or pore density) varies to be 10 pores per inch (PPI), 20 PPI and 30 PPI. It should be noted that the ligaments of the present porous cylinders are hollow associated with their fabrication process. All the test cylinders have the same dimensions such as the diameter of D = 42 mm, the length (of span) of l = 45 mm. Therefore, the span-to-diameter (i.e., the aspect ratio) is 1.07, classified as a short cylinder. A blockage ratio (diameter-to-rectangular flow channel width, D/W) is 0.105.
2.2. Endwall surface flow visualization using an oil-dye mixture technique Endwall surface flow pattern was visualized using an oil-dye mixture technique in the wind-tunnel. Florescent powder was mixed with a light diesel fuel, and the mixture was then applied to the cylinder surface. When the aerodynamic shear stresses redistribute the mixture, visualization of the surface flow patterns is realized. Before the application of the mixture, the surface is
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painted in black to enhance the reflectivity when using ultraviolet light to illuminate the surface flow pattern while it is being photographed. More details associated with this flow visualization technique can be found in Ref. [27]. 2.3. Pneumatic pressure and velocity measurements Static pressure variation along the plane of symmetry was measured by pressure tappings. Steel tubes with an inner diameter of 0.6 mm were inserted in the drilled holes along the plane of symmetry. Each tube was then connected to a differential pressure transducer (DSATM, Scanivalve Inc.). Axial velocity component of the mainstream was traversed to quantify boundary layer (displacement) thickness at x/D = ()4.5 using a total pressure probe with a flattened tip which allows pressure at 0.23 mm away from the endwall to be measured (the closest data point). The origin of the cylinder sample coincides with xaxis (i.e., x = 0). A Pitot tube is positioned at x/D = 2.5 upstream of the cylinder axis and at the mid-span and mid-width of the flow channel to monitor the mainstream’s centerline velocity. The Reynolds number based on the cylinder diameter D and centerline velocity was fixed at ReD = 1.0 104 (a sub-critical flow regime). The pneumatic pressure tappings and stagnation pressure probe are connected to the differential pressure transducer which sends data via a TCP/IP protocol to a data acquisition personal computer. 2.4. Endwall surface temperature mapping using an infrared thermography A constant heat flux is generated by a film heating element (A = 0.3 m 0.3 m) that is attached to the outer surface of the substrate. A film heat flux sensor is installed to measure a portion of the generated heat flux by the film heating element that leaves through the outer surface of the substrate i.e., q00 loss. The net heat flux (q00 ) used to calculate heat transfer coefficient (h) is estimated as:
Fig. 4. Photograph and schematic of the test rig showing a suction-type wind-tunnel with a contraction and a test section accommodating a short single cylinder.
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Fig. 5. Tested circular cylinders: (a) Impermeable (solid) cylinder, (b) 30 PPI porous cylinder (e = 0.94), (c) 20 PPI porous cylinder (e = 0.94), (d) 10 PPI porous cylinder (e = 0.94).
q00net ¼
Q in q00loss A
ð1Þ
where Qin the total input power calculated from the measured input current and voltage to the film heater and A is the heating area. The input power (Qin) was kept constant during the entire study e.g., 18 W (or q00 in = 200 W/m2). The heat loss (q00 loss) at this set Reynolds number was measured to be approximately 5.0% of the total input heat flux. Local temperature distribution was mapped using an infrared (IR) thermal camera (FLIR T640). A polymer based upper substrate is used as a viewing window for the IR camera. The infrared (IR) window used in the present study was made from a plastic (Edmund OpticsTM), had 60% of transmittance for the 8–14 lm wavelength and its thickness was 0.45 mm. For the current setup, a drop in radiation intensity across the window was observed. To account for the intensity attenuation across the window, a series of calibrations were carried out. A systematic intensity attenuation that results in the temperature drop of 6.0 K is obtained in the present Reynolds number. The surface temperature data only along the plane of symmetry was extracted to calculate local Nusselt numbers as:
Nuðx; 0; 0Þ ¼
hðx; 0; 0Þ kf =D
ð2Þ
where kf is the thermal conductivity of air evaluated at a film temperature ((Tin + Tm)/2) and the convective heat transfer coefficient along the plane of symmetry, h(x,0,0) was calculated as:
hðx; 0; 0Þ ¼
q00net ðT s ðx; 0; 0Þ T in Þ
the value used, an uncertainty in kf was calculated to be ±3%. The uncertainty in the cylinder diameter (D) was ±0.05%. The uncertainty in the Nusselt number was, therefore, calculated to be within ±4.0%. 2.5. Data reduction parameters and measurement uncertainty The Reynolds number ReD is defined based on the cylinder diameter D and the centerline axial velocity Uc measured at the mid-height of the flow channel with no pressure gradient as:
ReD ¼
qU c D l
ð4Þ
where l and q are the viscosity and density of air, respectively. Local static pressure along the plane of symmetry is evaluated using the wall pressure coefficient defined as:
Cp ¼
pðx=DÞ pref
qU 2c =2
ð5Þ
where p(x/D) and pref are the measured static pressure and reference pressure (at x/D = ()4.5) and Uc is the centerline velocity measured at x/D = ()4.5. Uncertainty associated with the Reynolds number and the wall pressure coefficient are estimated following a method reported by Coleman and Steele [29] (based on 20:1 odds) and found to be within 1.8% and 1.8%, respectively. Note that only random errors are considered here. It is assumed that the bias (systematic) errors could be minimized by careful calibrations.
ð3Þ
where Tin is the temperature of the mainstream measured at x/ D = 2.5 upstream of the cylinder axis by a bead T-type thermocouple and Tm is an area averaged surface temperature from the IR image. The experimental uncertainty of the Nusselt number was estimated using a method reported by Holman and Gajda [28] based on 20:1 odds. For heat flux measurement, the error associated with the voltmeter scanner was estimated to be less than ±5 lV, resulting in an error less than ±6 W/m2 for heat flux. The resolution of the temperature readings from the temperature scanner was found to be ±0.1 K for each thermocouple. This same resolution was applied to the infrared camera as the thermocouples were used in its calibration. The uncertainty in the heat transfer coefficient (h) was calculated from these values to be ±0.42 W/(m2 K). The thermal conductivity of air (kf) was averaged along the temperature range. Based on the difference between the actual value and
3. Thermal flows around an impermeable short cylinder (reference case) A reference case for thermal flows around a short circular cylinder was established for a Reynolds number of ReD = 1.0 104. The endwall flow visualization and heat transfer measurements were conducted. 3.1. Endwall flow pattern The surface flow pattern around a short cylinder, which is impermeable, is shown in Fig. 6, depicting a well-known horseshoe vortex system clearly marked by oil-dye mixtures. It is expected that horseshoe vortex system would be in the turbulent regime at the chosen Reynolds number. Since only the time averaged behavior of the horseshoe vortex system is captured, the flow pattern (horseshoe vortex) is symmetric with respect to the x-axis.
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Fig. 7. Temperature distribution around an impermeable short cylinder at ReD = 1.0 104.
Time averaging also makes the identification of flow features such as attachment points A0, A1 and A2 as well as the separation points S1 and S2, as seen in Fig. 1(b), difficult. The overall endwall surface flow pattern in Fig. 6 consistently shows topological features previously reported [2,23]. A separation point appears to be located at about x/D = 1.2 upstream from the cylinder axis (labeled as ‘‘S”). This point lies close to the position predicted by Baker [3] that the boundary layer separates from the endwall at x/D = ()1.12. In this photograph, the separation lines are indicated by brighter lines as a result of accumulated dye particles. The two legs of the horseshoe vortex merge at approximately x/ D = 1.75 downstream, forming a saddle point (SP5). Another saddle point (SP4) is observed immediately behind the cylinder approximately at x/D = (+)0.75. It may appear as a nodal point as indicated by the accumulated oil-dye particles, configuring an isolated island along the cylinder axis. It was observed, however, during the test that two distinct islands of oil-dye formed apart with respect to the x-axis but once the fluid stream was stopped these two islands melded together. The accumulation of the oil-dye corresponds to the high region of flow entrainment in Wake Region II, shown in Fig. 3. The above features observed in the present reference case (in Fig. 6) are topologically consistent with those illustrated in Fig. 3.
4.1. The alteration of basic fluidic features The alteration of fluidic features around a porous cylinder due to the mainstream that partially leaks through the cylinder, forming a ‘‘through-flow” can be indicated by some upstream flow properties such as an upstream velocity profile (or boundary layer thickness) and adverse pressure gradient. Fig. 8(a) shows velocity profiles along the y-axis (channel height) at x/D = ()4.5 on the plane of symmetry where velocities are normalized by the centerline velocity Uc (i.e., Uy/H = 0.5) for the empty channel case. Data from only a lower half of the channel
Impermeable cylinder Permeable cylinder, 20 PPI Empty channel x
0.6 x
0.5 0.4
y/H
Fig. 6. Endwall flow pattern and temperature distribution for an impermeable copper short cylinder at ReD = 1.0 104.
of 30 PPI, 20 PPI and 10 PPI. Furthermore a comparison of the basic fluidic features upstream of the impermeable and porous cylinders were made.
0.3 0.2 0.1 0x 0
0.2
0.4
3.2. Endwall heat transfer
Comparisons to the reference case were made through the use of endwall flow visualization and heat transfer data for fully permeable cylinders with the fixed porosity of 0.94 and pore densities
0.8
1
(a) 1
Leading edge
Impermeable cylinder Permeable cylinder, 20 PPI 0.8 Adverse pressure Skin friction gradient 0.6 0.4 0.2 0 -6
4. Thermal flows around porous short circular cylinders
x x
Cp
The temperature distribution on the endwall captured by the pre-calibrated infrared thermal camera is shown Fig. 7. Black strips that appeared in the IR image are the steel rods that support the IR window film. Since local vortex flow features act to enhance local heat transfer, there is a direct link whose comparison can be made. The local temperature in the regions influenced by the horseshoe vortex such as that downstream of the separation point, S, and around the cylinder’s vertex is substantially lower than the rest region. Near the saddle point, SP4, and Wake Region II, the local temperature is higher than the other regions, indicating a lower heat transfer region.
x
0.6 u/Uc
x x x x x x x x x x x x x x x x x x x x x x xx xx xx x xx xx xx xx x xx xx
-5
-4
-3 x/D
-2
-1
0
(b) Fig. 8. Approaching flow properties: (a) velocity profiles at x/D = ()4.5 measured from the cylinder axis, (b) stream-wise pressure gradients.
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height are displayed since the velocity profile was confirmed to be symmetric with respect to the mid-span of the cylinder in the x–z plane. The centerline velocity Uc in the plane of symmetry decreases as the flow is decelerated by the presence of the cylinder and finally stagnates on the cylinder. However, the overall velocity profile appears to remain unchanged. To detail the changes, the displacement thickness of the developed boundary layer (d⁄) was calculated based on the data in Fig. 8 (a) as:
d ¼
Z 0
H=2
uðyÞ dy 1 Uc
ð6Þ
The results are listed in Table 1. In comparison with the solid (impermeable) cylinder, the porous cylinder gives rise to a slightly thinner displacement thickness, which is, however, thicker than that of the empty channel. The pressure distribution upstream of the cylinder until the cylinder’s leading edge in the plane of symmetry is plotted in Fig. 8(b). Before x/D = ()4.5, the static pressure is decreased although its extent is marginal. This decrease in the static pressure is due to pressure loss resulting from skin friction. Farther downstream from which, the Cp value begins to monotonically and significantly increase up to the nearest measurement point at x/D = ()0.75 (from the axis of the cylinder). This pressure rise (or adverse pressure gradient) is mainly due to the stagnation of the mainstream on the cylinder. With the porous cylinder having the pore density of 20 PPI, a similar pressure variation is observed along the plane of symmetry. The only difference is a reduced adverse pressure gradient from the upstream location at x/D = ()2.0 until the leading edge of the cylinder (x/D = ()0.5). 4.2. Endwall thermal flow patterns The slight changes in the boundary layer profile and adverse pressure gradient discussed with Fig. 8 may influence horseshoe vortices to be formed around the porous cylinders. Furthermore, a portion of the mainstream enters the porous cylinder forming a through-flow, and is expected to cause the alteration of the horseshoe vortices and wakes. Fig. 9 exhibits the endwall flow pattern visualized for the three fully permeable (porous) cylinders: 30 PPI (Fig. 9(a)), 20 PPI (Fig. 9 (b)), and 10 PPI (Fig. 9(c)). A common noticeable endwall flow pattern is the shortening of the distance between the separation point (S) and the leading edge of the cylinder. For the impermeable cylinder, the point at which the boundary layer separated from the endwall (S) was identified at x/D = ()1.2. For the 30 PPI porous cylinder, the separation point was identified at x/D = ()1.0. The allowance for the through-flow by the porous cylinder would decrease the adverse pressure gradient upstream of the cylinder, thereby causing the boundary layer separation to be delayed. For the 20 PPI and 10 PPI porous cylinders, the separation points were identified at x/D = ()0.9 and x/D = ()0.75, respectively. A lower pore density results in a larger decrease in the adverse gradient, and subsequent further delay in the separation of the boundary layer. Furthermore, the low shear stress region immediately downstream of the cylinder is elongated. A further elongation occurs as the pore density is decreased (i.e., the pore size becomes bigger).
Table 1 Boundary layer displacement thickness calculated at x/D = ()4.5. Cylinder sample
d⁄ [mm]
D/d⁄
Impermeable (solid) cylinder Permeable cylinder, 20 PPI Empty channel
1.81 1.78 1.67
23.2 23.6 25.2
Fig. 9. Endwall flow pattern for permeable short cylinders at ReD = 1.0 104: (a) for 30 PPI, (b) for 20 PPI, (c) for 10 PPI.
The horseshoe vortices enhance endwall heat transfer especially at the fore cylinder part as seen by a low temperature region there in Fig. 7. The overall endwall temperature distribution is similar to the endwall flow patterns in Fig. 6 for the impermeable cylinder. With the porous cylinder of 30 PPI (Fig. 10(a)), the variation of local temperature at the fore region of the porous cylinder is not distinguishable although that at the aft cylinder is marked by a high temperature region. However, with the porous cylinder of 20 PPI (Fig. 10(b)) and 10 PPI (Fig. 10(c)), the thinning of the low temperature region upstream of the cylinder becomes obvious. The elongated high temperature regions behind the cylinder coincide with the low shear stress region in Fig. 9. 4.3. Wakes behind the porous cylinders Before identifying the thermo-fluidic effects the through-flow might have on endwall heat transfer, further fluidic evidence of the through-flow needs to be understood.
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x/D = 3.0
x/D = 5.0
z/D
2.5 2 1.5 1 0.5 0 -0.5
0 u/Uc 1
0
1
(a) x/D = 3.0
x/D = 5.0
z/D
2.5 2 1.5 1 0.5 0 -0.5
0 u/Uc 1
0
1
(b) x/D = 3.0
x/D = 5.0
z/D
2.5 2 1.5 1 0.5 0 -0.5
0 u/Uc 1
0
1
(c) x/D = 3.0
x/D = 5.0
z/D
2.5 2 1.5 1 0.5 0 -0.5
0 u/Uc 1
0
1
(d) Fig. 11. Wake velocity profiles measured at x/D = (+)3.0 and x/D = (+)5.0 (downstream) from the cylinder axis: (a) solid cylinder, (b) porous cylinder with 30 PPI, (c) porous cylinder with 20 PPI, (d) porous cylinder with 10 PPI.
Fig. 10. Temperature distribution around permeable short cylinders: (a) for 30 PPI, (b) for 20 PPI, (c) for 10 PPI.
The wake profile measured at the downstream position, x/ D = 5.0, shows that both the impermeable cylinder and the porous cylinder with 30 PPI have a similar profile (Fig. 11(a, b)). Slightly upstream at x/D = (+)3.0 (closer to the aft cylinder), the profile for the impermeable cylinder is still approximately the same; the downstream wake having mostly re-established following the disturbance by the cylinder. For the 30 PPI porous cylinder, however, there is a marked change at x/D = (+)3.0, the velocity at the cylinder axis being approximately half of that at x/D = (+)5.0; the effects of the porous cylinder on the downstream wake last longer. Based on the wake profile data in Fig. 11(a, b), fluidic sketches are presented in Fig. 12(a, b). A significantly different profile is measured behind the porous cylinder with 20 PPI and 10 PPI at both downstream positions. At x/D = (+)3.0, erratic negative pressure readings were recorded close to the cylinder axis signifying that the fluid stream was moving upstream – the reverse flow. This would be because of the presence of vortices in the cylinder wake, as noted by Sahin [21]; these vor-
tices having moved further downstream for the lower pore density porous cylinder. At x/D = (+)5.0, there is no reverse flow observed; the axial velocity magnitude being lower for the 10 PPI porous cylinder than for the 20 PPI porous cylinder. The lower density porous cylinder results in the wake taking longer to re-establish as illustrated in Fig. 12(c, d), which is consistent with the observation made by Khashehchi et al. [26]. In summary, the wake profiles at the mid-span of the cylinder for the impermeable and porous cylinders are observed to be similar to those on the endwall: the through-flow within the porous cylinder resulting in a delay of the wake re-establishment with a lower pore density resulting in a longer delay. 4.4. Flow resistance in the porous cylinder It has been observed that the mainstream that partially leaks through the porous cylinder forming a through-flow causes: (a) the thinning of locally enhanced heat transfer region in front of the cylinder’s leading edge, and (b) an enlarged ‘‘thermally” less active region behind the cylinder and whose longitudinal coverage is elongated as more flow leaks through the porous cylinder.
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The permeability was then calculated from the modified Darcy equation [30] as,
dp 1 1 C F qu ¼ þ pffiffiffiffi dx lu K K l
ð7Þ
where dp/dx is the pressure drop per unit length along the nominal flow direction, CF is the inertial coefficient, u is the mean flow velocity. By plotting the quantity (dp/dx)(1/lu) as a function of qu/l for the range of flow rates considered, the two parameters are calculated graphically as shown in Fig. 13 where permeability is the intercept and the inertial coefficient is the slope. Results are summarized in Table 2. Higher permeability that implicitly indicates lower flow resistance is obtainable from a porous sample with bigger pore diameter for a given porosity. 4.5. Through-flow vs. wakes
Fig. 12. Sketch of flow patterns around the cylinder; (a) solid cylinder, (b) porous cylinder with 30 PPI, (c) porous cylinder with 20 PPI, (d) porous cylinder with 10 PPI.
The amount of the through-flow within the porous cylinder varies with the pore density. Within the porous cylinder, the flow resistance exists and is proportional to the thickness of the porous cylinder. Therefore, the highest resistance is encountered along the plane of symmetry. As a result, the mainstream that enters tends to detour to exit the porous cylinder via a flow path with less flow resistance. By the time when the through-flow exits the porous cylinder, the momentum is substantially dissipated. As the pore density becomes lower, the more through-flow can penetrate the porous cylinder due to a less flow resistance within the porous cylinder. The through-flow within the porous cylinder then becomes more straight or parallel to the x-axis. Even for a porous medium at a fixed porosity, pressure drop (or flow resistance) within the porous medium varies with the pore density. For the present three pore density samples, permeability (K) that is an imaginary length scale but an inversely and implicitly indicative of flow resistance was calculated from the pressure drop data as a function of mean (or nominal) flow velocity. In the parallel flow configuration, the pressure drop of the three porous samples was measured prior to the present study. A higher pressure drop is obtained from a porous sample with a smaller pore size for a fixed porosity (e = 0.94). Topologically, a porous sample with a smaller pore size has thinner ligaments.
The flow momentum that is introduced into the cylinder wake after a substantial dissipation within the porous cylinder at a certain exit angle, delays the horseshoe vortices from folding into form the downstream flow pattern as seen in Fig. 3. This delay moves the point at which the mainstream merges downstream (SP5) and moves the vortex VD and saddle point SP4 downstream (the horseshoe vortices being absorbed into VD). The downstream movement of these vortices means that Wake Region I is lengthened. This region having very little fresh fluid entrainment between the cylinder surface and SP4 (referring to Fig. 3). This region in Fig. 9(a–c) was identified by the lack of any significant disturbance to the oil-dye by the endwall shear stresses. The lower the pore density, the lower the resistance of that is encountered by the through-flow, the higher the momentum of the through-flow exiting the cylinder and subsequently the further downstream the saddle points SP4 and SP5. For the 30 PPI porous cylinder, Fig. 9(a), the size of the pooled island of oil-dye downstream of the cylinder is of a relatively similar size as that for the impermeable cylinder as seen in Fig. 6. As the pore density is increased, a notable increase in the size of the pooled oil-dye is observed. It is unlikely that this is due to a strengthening of Wake Region II with a decrease in the pore density but rather because of the increased distance between Wake Region II and the cylinder surface. The larger the distance, the more oil-dye that is carried towards the region of high fluid entrainment. As stated previously, it was observed during the tests that two distinct islands of oil-dye formed in Wake Region II but once the fluid stream stopped, these two islands melded together. Fig. 9(c)
Fig. 13. Graphical deduction of permeability (K) and inertial coefficient (CF) from the measured pressure drop (Dp/L) and mean velocity (u) for three copper porous samples (e = 0.94).
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Table 2 Properties of the copper porous samples where ks = 398 W/(m K) where the surface area density (or surface-to-volume ratio) asfof the porous samples was obtained according to the method used by Fuller et al. [31] based on the cubic model since the ligaments are hollow. Porosity, e Pores per inch (PPI) Pore diameter, dp[mm] Surface area density, asf[m2/m3] Permeability, K [m2]107 K1/2 [m] 104 Inertial coefficient, CF
0.94 (measured) 10 20 2.54 1.27 2286 5370 3.63 1.45 6.02 3.81 0.115 0.0848
30 0.85 8009 0.712 2.67 0.0952
still shows this to a small degree on the downstream side of the pooled dye.
Fig. 15. Schematic of the downstream flow topology on the endwall and trend in Nusselt number.
4.6. Wakes vs. endwall heat transfer To quantify how the through-flow alters local surface temperature distribution (or endwall heat transfer), the temperature data was extracted from Figs. 7 and 10 along the plane of symmetry. The results presented in the Nusselt number are plotted in Fig. 14 and show that local heat transfer is decreased as approaching the leading edge (LE) of the cylinder but turns to increase for all four included cases. Their turning point appears to be associated with the separation point (S). From the impermeable cylinder to porous cylinders (30 PPI ? 20 PPI ? 10 PPI), the turning point moves slightly towards the LE. After each turning point, local heat transfer increases steeply, having the highest Nusselt number obtained from the impermeable cylinder whereas the lowest heat transfer is obtainable from the porous cylinder with 10 PPI. A dramatic difference can be seen on local heat transfer downstream of the cylinder. Substantially low local heat transfer take place behind the cylinder that is porous. The trend of the Nusselt number for the impermeable cylinder (Fig. 14) exhibits a minimum point at a position of x/D = (+)0.75. Comparing this to the endwall flow pattern (Fig. 6), this position approximately coincides with the identified position of SP4. A similar minimum point is seen for all three porous cylinder cases (30 PPI, 20 PPI and 10 PPI). However, their positions are upstream of saddle point SP4, within Wake Region I (Fig. 3). After the minimum point the Nusselt number, for the impermeable cylinder, increases before reaching a maximum at x/D = (+)1.5. Comparing to the endwall flow pattern in Fig. 6, this maximum point occurs at the identified position of the saddle point SP5. The Nusselt numbers for the porous cylinders also exhibit an upward trend after the minimum point, the data does not
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Impermeable Cu cylinder Porous Cu cylinder, 30 PPI Porous Cu cylinder, 20 PPI Porous Cu cylinder, 10 PPI
90
NuD
80 70 60 50 40 -1.5
-1
-0.5
0
0.5
x/D
1
1.5
2
2.5
Fig. 14. Nusselt numbers along the x-axis in the plane of symmetry for the solid cylinder (as reference) and Cu porous with three pore densities.
reach a maximum value but is assumed to follow a similar trend as that for the impermeable cylinder; the saddle points SP5 for the porous cylinders are all outside the region that temperature measurements were made. A schematic of the flow topology on the endwall and trend in the Nusselt number for the porous cylinders is shown in Fig. 15. In an area of high fluid entrainment from the core flow, such as Wake Region II (Fig. 3), there will be a higher degree of heat transfer at the endwall. Wake Region II (as per Fig. 3) and is noted as having high levels of fluid entrainment. Greater heat transfer would be expected at SP5 as there is both upstream flow, towards Wake Region II, and downstream flow, following the mainstream. This is why there is a local maximum seen in Fig. 15 for the impermeable cylinder, at SP5. In an area of low fluid entrainment (Wake Region I), the fluid stream is trapped meaning that there will be a decrease in the heat transfer. The minimum heat transfer would then occur somewhere this region. For the impermeable cylinder, the point of minimum heat transfer occurs at the downstream side of Wake Region I, close to SP4 (not coinciding with SP4). The minimum heat transfer for the 30 PPI cylinder occurs at approximately x/D = 0.6 (Fig. 14), placing it closer to the cylinder circumference than SP4, further upstream than that of the permeable cylinder. As the pore density is decreased, the point of minimum heat transfer moves downstream. For the 10 PPI porous cylinder, the minimum is closer to SP4 than the cylinder circumference. 5. Conclusions A series of experiments to investigate the alteration of horseshoe vortices formed around a short circular cylinder due to the mainstream that is partially leaked through the cylinder, have been performed at a turbulent Reynolds number. The findings in this study are summarized as follows. (1) The presence of the mainstream that is partly leaked through the porous cylinder causes a delay of the separation point of the incoming boundary layer flow at the fore cylinder and the elongation of the wakes at the aft cylinder. An adverse pressure gradient along the plane of symmetry is reduced as well. (2) The amount of through-flow is determined by the permeability (or pressure drop): a lower pore density porous cylinder for a fixed porosity allows more through-flow, leading to the more greatly elongated wakes. The flow that loses its substantial momentum within the cylinder meets the external mainstream at a certain exit angle when leaving the cylinder (blowing effect), causes the elongated wakes.
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(3) In the elongated wake regions, substantially lower endwall heat transfer takes place, posing thermally less active regions since the lower shear stresses exist there. (4) Before the inner saddle point (SP4) on the endwall, a local minimum of heat transfer exists whereas a local maximum endwall heat transfer coincides with the outer saddle point (SP5) along the plane of symmetry. (5) Regarding the endwall heat transfer modification, the presence of the through-flow within the cylinder (or an object) acts negatively by (a) narrowing a thermally active region under the influence of horseshoe vortices at the fore cylinder and (b) elongating thermally less active regions at the aft cylinder.
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