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On the thermal and structural characteristics of an artificially generated young turbulent spot
International Journal of Heat and Mass Transfer 92 (2016) 850–858
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On the thermal and structural characteristics of an artificially generated young turbulent spot Weerachai Chaiworapuek ⇑, Chawalit Kittichaikarn Department of Mechanical Engineering, Faculty of Engineering, Kasetsart University, Bangkok 10900, Thailand
a r t i c l e
i n f o
Article history: Received 8 February 2015 Received in revised form 20 August 2015 Accepted 17 September 2015 Available online 1 October 2015 Keywords: Turbulent spot Boundary layer transition Liquid crystals Heat transfer
a b s t r a c t The unsteady heat transfer from an isothermal wall to a water flow via a turbulent spot at an early stage was investigated experimentally using thermochromic liquid crystals. With a Reynolds number of approximately 75,000 at the test area, an artificially generated turbulent spot was induced by a water injection in a low-turbulence water tunnel with a turbulent intensity of 0.93%. The ratio of the spot heat transfer rate to the heat transfer rate existing without the spot was determined to evaluate the turbulent spot effectiveness. Selected images of both the Nusselt number and the spot effectiveness were illustrated to visualize the spot development. Interesting results were obtained soon after the spot initiation, as the spot provided an averaged heat transfer up to 10% above the laminar state. The results also indicated a significant influence of the water injection consequent to the non-linear behavior during the young state of the turbulent spot. The turning point from young to mature state was reported. Moreover, the spot footprint appeared as a hand-like shape and propagated downstream with 38% of the free stream velocity. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Turbulent spots occur under boundary layer transitions from laminar to turbulent flow and are responsible for the change in boundary layer properties and profile observed between the laminar and the turbulent state. The turbulent spot was first discovered by Emmons [1], who noticed a small turbulent patch on a water table. Emmons also showed that the turbulent spot could be generated directly by releasing a water droplet to strike the target surface. Later, Schubauer and Klebanoff [2] undertook experiments using a hot wire anemometer and found that the turbulent spot had an arrowhead shape, which propagated downstream with 88% and 50% of the free stream velocity at its leading and trailing edges, respectively. These researchers also found a stable state region similar to the laminar flow following the passage of the turbulent spot. This area was highly stable, and no breaking down was likely to appear. This region was ‘‘the recovery trail” or ‘‘the calmed region”. The results from many contributions [3–9] have also confirmed that the turbulent spot body has an arrowhead-like structure. Using a hot wire anemometer, Wygnanski et al. [4] suggested that the entire turbulent spot, based on the ensemble averaged analysis, ⇑ Corresponding author. E-mail address: [email protected] (W. Chaiworapuek). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.09.053 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
was a single large vortex. However, multiple sub-structures have been detected by different measurement techniques [5,8–13]. The growth mechanism of the turbulent spot in the streamwise and heightwise directions is due to the gulping entrainment process at its front interface and the nibbling process at its trailing edge [7]. Meanwhile, destabilization is considered the primary cause of the lateral growth of the turbulent spot [7,12]. In the bypass transition process, the turbulent spot can be directly initiated by injecting fluid into the flow or increasing the level of the free stream turbulence. Haidari and Smith [12] suggested that the initial hairpin vortex generated by the injection process was the primary vortex. This vortex then induces the subsidiary vortex and the secondary vortices, which were also considered main features of the turbulent spot. However, the spot would take some time to develop to a linear or mature state. This time becomes shorter with increasing Reynolds number [4]. The thermal structure of the turbulent spot was first investigated using a cold wire by Van Atta and Helland [18]. When it was compared with the contour of velocity perturbation, yielded by Zilberman et al. [19], the temperature and velocity fields were found to be strongly anti-correlated. Van Atta and Helland [18], Antonia et al. [20], and Chong and Zhong [21] were all in agreement that the near wall temperature inside the turbulent spot was below the local temperature before the spot arrival. This finding is consistent with the results reported by Zhong et al. [22],
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Nomenclature As CLE Cm CTE Ein Eg Est H hx k Nux Nu Pr Q Qns Qs q q qns qs Rex T t Tal
turbulent spot area (m2) turbulent spot velocity at leading edge propagation rate of the turbulent spot centroid turbulent spot velocity at trailing edge energy inflow (W) energy source (W) energy storage (W) Hue signal value heat transfer coefficient (W/m2 °C) thermal conductivity (W/m °C) local Nusselts number average Nusselts number Prandtl number rate of heat transfer (W) heat transfer rate of heated plate through the area of turbulent spot (W) rate of heat transfer through turbulent spot (W) heat flux (W/m2) average heat flux (W/m2) heat flux of heated plate through the area of turbulent spot (W) heat flux through turbulent spot (W) local Reynolds number temperature (°C) time, counted after spot initiation (s) temperature on aluminum plate (°C)
Sabatino and Smith [23], Chaiworapuek et al. [24], who employed the technique of temperature mapping using thermochromic liquid crystals. Typically, the shape of the turbulent spot’s footprint can be directly examined via the region of heat transfer [22]. Its foremost edge is defined as the leading edge, as shown in Fig. 1. The trailing edge and the half spreading angle can also be obtained as shown in the figure. The footprint underneath the turbulent spot propagates downstream with velocities in the range of 74–83% and 53–57% of the free stream velocity at the leading and trailing edges, respectively [16,17,25]. Additionally, the spreading half angle of the spot footprint was reported to be between 4° and 6.8° [14–17,25]. Chaiworapuek et al. [17] found that the arrowhead-like shape was first found when the leading edge reached the streamwise distance of x/d⁄ = 94.54 from the location of the spot generator. Before that, a hand-like structure was observed [17]. Sabatino and Smith [23] found that a peak in heat transfer at approximately 15% above the laminar condition occurred in the calmed region. This peak is caused by the cooler fluid entering from the upstream higher level, following Johnson [26].
Flow Direction
Wing tip
Half spreading angle Locus of wing tips
Virtual origin
Ta U Ua W X x x0 Dx Y y Dy Dz
free stream temperature (°C) velocity (m/s) free stream velocity (m/s) turbulent spot half width (m) dimensionless streamwise distance streamwise distance (m) streamwise location of the spot generator (m) streamwise length of each pixel (m) dimensionless spanwise distance spanwise distance (m) spanwise length of each pixel (m) heightwise length of each pixel (m)
Greek symbols a half spreading angle (°) dL laminar boundary layer thickness (m) d boundary layer displacement thickness at the spot generator (m) s dimensionless time es turbulent spot effectiveness es average turbulent spot effectiveness n unheated starting length (m) r non-dimensional spot propagation parameter
Free stream flow
Spot generator
PVC sheet
Bleeding channel Strip heaters Thermal insulator Supporter Pressure system
Steel plate
Fig. 2. Schematic of the test section in the low free stream turbulence water tunnel.
Hence, this study experimentally investigates the thermal and structural characteristics of a turbulent spot footprint. The test was conducted in a low free stream turbulence water tunnel with a Reynolds number of approximately 75,000 at the test area. To obtain all spot parameters, including the Nusselt number, the heat flux, the spot effectiveness, the velocities, the half spreading angle, and the non-dimensional spot propagation parameter, the thermochromic liquid crystals are used together with an analytical solution derived from the energy balance, following Chaiworapuek et al. [24]. All presented results provide valuable insight into the growth and heat transfer process of the turbulent spot footprint under the bypass transition.
Leading edge
Calmed region Trailing edge
Fig. 1. Schematic of a turbulent spot on a flat plate.
2. Experimental setup Fig. 2 shows a schematic diagram of the test section of the closed-loop low free stream turbulence water tunnel used in this
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study. The test section of 0.15 0.2 m was made from 10 mm thickness perspex to gain clear visibility throughout the experiment. The turbulent spot was directly initiated by injecting water through a 1 mm diameter hole, connecting with the pressure system at the distance of x0 = 0.3 m or x/d⁄ = 94.54 from the leading edge of the flat plate. At this point, the boundary layer displacement thickness d⁄ was measured as 3.17 mm using a Dantec fiber film probe type 55R11. A bleeding channel was located at the most upstream point of the section. The bleeding rate was adjustable by the gate valve, installed at an exit to provide a suitable condition for a new boundary layer, refreshed at the leading edge. The flat plate was a 3 mm thick aluminum plate for the low weight and good thermal conductivity. The nine strip heaters were set up below the aluminum plate to generate the isothermal condition of the flat plate. They were controlled using proportional-inte gral-derivative (PID) controllers that received local input signals from three stations of type-K thin leaf thermocouples. The thermocouples were finely glued on the top surface at 40, 58, and 78 cm from the leading edge. To prevent unwanted thermal loss, a high thermal resistance plate was laid beneath each heater. All strip heaters and the insulator were supported by a supporter and pressurized by a 3 mm thick steel plate at the lowest layer. On the top layer, a black polyvinyl chloride sheet with a thickness of 120 lm was well bonded with the aluminum plate. The plastic sheet had a thermal conductivity much lower than the aluminum plate; consequently, the heat transfer from the heaters was retarded and damped by the polyvinyl chloride sheet when conducted to an adjacent interface. Using the dimmer connected to each heater, power control of all heaters was performed independently, and a continuously isothermal condition of the flat plate was obtained. Temperature uniformity of the heated surface was confirmed by the uniform color of the liquid crystals coating the whole plate. Constant pressure of the spot generator was provided by the pressure system as shown in Fig. 3. A water tank was used to reserve the water for the injecting system. To control the pressure level of the injector, an air pump must fill a diaphragm tank with pressured air to obtain a suitable pressure level for the main system. The water injected from the spot generator was directly released through a solenoid valve with the duration of 0.02 s, activated by a programmable logic controller (PLC) board. Thus, this pressure level of the injected water was held constant through the experiment. In this study, the thermal footprint of the turbulent spot was qualitatively and quantitatively revealed by thermochromic liquid crystals coated at a thickness of approximately 20 lm on top of the plastic sheet. The liquid crystals used in the experiment were micro encapsulated Chiral Nematic crystals with an active range from 27 °C to 29 °C. They were mixed with CC300 binder to increase the water resistance. After coating, a continuous thin layer of clear vanish was applied to resist the degradation of the
Spot generator
P
reflective property of the liquid crystals by the water. Two 70 watt fluorescent bulbs, each with a diameter of 2.5 cm, were assembled beside the test section. Their white light, containing the full spectrum of wavelengths from red to blue, was used to reflect the colors from the crystals. Additionally, glossy reflectors were installed with each bulb to boost the light intensity. To acquire the image of the turbulent spot footprint, a video camera with a frame rate of 24 frames per second was installed 1.5 m above the test plate. At this height, the test area image of 0.31 m 0.12 m was obtained with a resolution of 281 pixels 681 pixels. The uniform flow was continuously supplied by a 2.5 HP centrifugal pump at the free stream velocity of Ua = 0.17 m s1, measured using a Nixon Streamflo velocity meter type 403 at the streamwise distance of x/d⁄ = 158 from the leading edge. This velocity corresponded to a Reynolds number of approximately 75,000 at the recorded area. At this position, a turbulence intensity of 0.93% was measured using a Dantec fiber film probe type 55R11. The velocity profile of the unheated laminar boundary layer was also obtained, as depicted in Fig. 4. All terms in the figure are defined in the Nomenclature section. 3. Mathematical analysis In the test section, the liquid crystals were fully coated on the heated wall for temperature measurement. The relationship between the scattered color and the temperature of the liquid crystals was established by the calibration process [17]. It was fit by the 3rd order polynomial equation as follows:
T ¼ 58:483H3 61:246H2 þ 23:156H þ 23:898
ð1Þ
It is noted that the scattered color is represented as H, which is the Hue signal from each pixel of the image. This equation gave a goodness of fit (R2) of 0.99873. In this study, the contour of the averaged temperature at each time step was used to construct the contours of the averaged heat transfer coefficient and averaged heat flux. It was calculated by pixel-by-pixel averaging of 30 individual turbulent spot footprints. Then, the heat transfer coefficient was determined from this averaged temperature contours using the energy balance method [24]. Initially, it was governed by the energy conservation equation as follows:
Ein þ Eg ¼ Est
ð2Þ
However, this method was used under the assumption that the temperature between the plastic sheet and the aluminum plate remains constant as the spot passes. Hence, with changing temperature, it was determined as follows:
Air pump
Diaphragm tank Air Water tank Water
Solenoid valve
P
Fig. 3. Schematic of the pressure system.
Fig. 4. Velocity profile, measured at the distance of x/d⁄ = 158 from the leading edge of the test plate.
W. Chaiworapuek, C. Kittichaikarn / International Journal of Heat and Mass Transfer 92 (2016) 850–858
T al ¼
Q Dz þT kðDxDyÞ
853
ð3Þ
Using this assumption, Q was evaluated based on the heat rate from the flat plate under the laminar flow, defined as follows:
Q ¼ hx ðDxDyÞðT T a Þ
ð4Þ
where hx is the local heat transfer coefficient on the flat plate with the unheated starting condition under laminar flow. Following Kays and Crawford [27], hx is given by
hx ¼
1=3 0:332Re1=2 k x Pr 3=4 1=3 x ½1 ðn=xÞ
ð5Þ
Fig. 6. The contour of the RGB signal at s = 0, 22.3, 33.5 and 44.6 after turbulent spot initiation.
Then, the local heat flux was simply calculated as
q ¼ hx ðT s T a Þ
ð6Þ
where q is the local heat flux from the flat plate to the water flow. Also, the local Nusselt number is defined as
hx x Nux ¼ k
ð7Þ
The local Nusselt number and heat flux obtained from the energy method are validated by comparison to the values determined from the theoretical formula (Eqs. (5) and (6)) as shown in Fig. 5. On the x-axis, the streamwise distance is presented in dimensionless form by comparison to the boundary layer displacement thickness at the spot generator as follows:
X ¼ ðx x0 Þ=d
ð8Þ
Both the Nusselt number and the heat flux are plotted through the distance, X, from 8 to 100, which is the area where the young turbulent spot occurs. It is found that the local Nusselt number and heat flux obtained from this study are consistent with the values achieved from theoretical prediction, which confirms the accuracy of the energy balance method. Moreover, the turbulent spot effectiveness, es, was determined from the ratio of the turbulent spot heat transfer rate, Qs, to the heat transfer rate existing without the spot, Qns. It was expressed as
es ¼
Qs q ¼ s Q ns qns
ð9Þ
Notice that this equation compares Qs and Qns through the area of the footprint. Thus, the rate of heat transfer in this study is alternately represented by the heat flux.
Fig. 5. The local Nusselt number and heat flux from the theoretical prediction and the experiment.
4. Results and discussion 4.1. Thermal analysis of the turbulent spot Fig. 6 presents the growth of the young turbulent spot after initiation beneath the main stream, which flows from left to right. The obtained colors were scattered by the liquid crystals. The streamwise distance on the x-axis starts from zero at the location of the spot generator and it is directly non-dimensionalized, as expressed previously. A spanwise distance on the y-axis was presented in dimensionless form as
Y ¼ y=d
ð10Þ
The zero distance is also set at the location of the spot generator. The value s is the dimensionless time, counted from the injection time. It is defined as the time in which the free stream water can move further with a streamwise distance of one boundary layer displacement thickness at the location of the spot generator, or
s¼
t d =U a
ð11Þ
The images in Fig. 6 were captured at s = 0, 22.3, 33.5, and 44.6 after spot initiation. The location of the spot generator was at the co-ordinate of X = 0 and Y = 0. At s = 0, the temperature of the test plate was kept at 28.3 °C shown in blue on the right side of all images in Fig. 6. The non-blue color, appearing from X = 0 to 8, showed the unheated starting zone. These images show the structure of the young turbulent spot after initiation by the spot generator. The arrival of the turbulent spot resulted in a color change from blue, as shown in the images at s = 0, 22.3, 33.5, and 44.6. It was found that the turbulent spot was first characterized by a 4-streak structure. The width of each streak was approximately two d⁄. Its two central streaks have very high heat transfer ability when the local Nusselt number is considered, as shown in Fig. 7. The streamwise and spanwise distances were newly established as X from 8 to 103 and Y from 18 to 18, respectively. This contour presents the local Nusselt number, evaluated by the color bar below at s from 44.6 to 200.9 with an interval of approximately 22. The blue shade appearing at X coordinates of 30 and 90 represents the location of the thin leaf thermocouples. Fig. 7 shows that the turbulent spot provides higher heat transfer ability than the unperturbed area beneath the laminar boundary layer. The area with the maximum Nusselt number first appeared as the two central streaks, quickly formed and propagating within the streamwise distance only X = 0–20. Immediately after the water injection, the first hairpin vortex formed from the injection process. This vortex brought cold water that was initially upstream of the spot generator to impact the heated surface following the flow direction of the vortex. The difference in temperature between the heated plate and the cold water was the cause of
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Fig. 7. The contour of the local Nusselt number at s = 44.6, 67, 89.3, 111.6, 133.9, 156.3, 178.6, and 200.9 after turbulent spot initiation.
the high-Nu region. However, the value of the maximum Nu decreased at s > 67. After the creation of the first or primary hairpin vortex, the subsidiary and secondary vortices were induced. The spot body was found to pull in all the surrounding water via the entrainment process, including the near wall hot water that initially relied on the heated surface. The colder water on the above layer was rapidly drawn towards the surface to replace the fluid that was now swept into the spot body. Hence, the process of heat transfer through the turbulent spot occurred. However, the influence of the temperature difference vanished completely by s > 200.9. At this time, it was found that the region with the maximum Nusselt number was newly located near the leading edge, inside the turbulent spot body. Notice that the maximum Nu at this point was approximately 360, which was lower than the maximum Nu occurring due to the first hairpin vortex. The maximum Nusselt number detected inside the footprint area is shown along the dimensionless time after spot initiation Fig. 8. The maximum Nusselt number increased to a maximum of 480 at s = 55 and was clearly located at the core of the two central streaks, as shown in Fig. 7. Then, the maximum Nusselt number decreased to 360 at s = 187. Furthermore, the average Nusselt number, Nu, through the spot footprint area, As, was also presented in each time step. It was evaluated by
Fig. 8. The maximum and average Nusselt numbers of the turbulent spot footprint.
Nus ¼
1 As
Z Nu dAs
ð12Þ
As the length of each image pixel was known, As in this study was determined by counting the pixel inside the turbulent spot footprint. The true shape of the footprint can be represented by the heat flux contour [16]. The boundary of the footprint was identified with the threshold of 10% of the maximum heat flux [28]. From the results, it was found that the average Nusselt number was highest at approximately 350 at s = 44.5. It then decreased to 270 and remained constant until s = 175. The structural behavior is further clarified by displaying the heat flux from the footprint. The contour of the local effectiveness of the turbulent spot, derived from the heat flux using Eq. (9), is depicted in Fig. 9. The main behavior of the thermal footprint from this contour is consistent with the behavior shown by the Nusselt number. The figure shows that the turbulent spot footprint first appeared as 4 streaks or a hand-like structure. It elongated in the streamwise direction at a higher rate than in the spanwise direction. The two central streaks transferred heat flux up to 25% above the unperturbed surface. Meanwhile, the local spot effectiveness at the two outer streaks was found to be approximately 20%. It is noted that the maximum effectiveness region occurred inside the two central streaks propagated within the distance and time of X = 20 and s = 89.3, respectively. Beyond this time, the magnitude of the heat transfer in this region decreased, and it later appeared as a part of the becalmed region located at the rear section of the footprint. Thus, the location of the maximum effectiveness jumps from the distance of X = 20 to 70 at the time of s = 111.6 to 133.9. This discontinuity shows that the effect of the first hairpin vortex, caused by the spot initiation, remained until s = 133.9. After this time, the region of maximum effectiveness appeared relatively near the leading edge of the footprint as it moved downstream with the turbulent spot. The maximum effectiveness was found at 25% above the laminar state again when the leading edge of the footprint reached X = 110. At s P 200.9, the value of the maximum effectiveness was still increasing as the spot convected further downstream. Fig. 10 shows the maximum and average heat fluxes of the turbulent spot footprint along the dimensionless time from 45 to 185 after the spot initiation. The maximum heat flux occurring inside
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Fig. 9. The contour of the local spot effectiveness at s = 44.6, 67, 89.3, 111.6, 133.9, 156.3, 178.6, and 200.9 after turbulent spot initiation.
Fig. 10. The maximum and average heat flux through the turbulent spot footprint.
the turbulent spot reached a peak of 690 W/m2 at s = 60 and decreased to a minimum of 220 W/m2 at s = 135. Unlike the Nusselt number, the maximum heat flux apparently increased again after s = 135. This result is in agreement with the average heat flux, q, which is determined from the relation
qs ¼
1 As
Z
q dAs
ð13Þ
The solid line that shows the average heat flux through the turbulent spot shows the same behavior as the maximum heat flux in Fig. 10. The peak was found to be approximately 240 W/m2 at s = 63. The minimum of 93 W/m2 was found at the same time with the dashed line and then slowly increased to 103 W/m2 at s = 185. To compare the amount of heat transfer through the turbulent spot with the heated surface under the laminar thermal boundary layer at the same spot area, the average spot effectiveness was proposed, as shown in Fig. 11. This parameter was evaluated as follows:
es ¼
1 As
Z
e dAs
ð14Þ
Fig. 11 shows that the characteristics of the average effectiveness are similar to the measured heat flux. After the spot initiation at s = 0, the perturbed area transferred heat from the heated sur-
Fig. 11. The average spot effectiveness through the dimensionless time from 45 to 185.
face to the water up to 10% above the laminar state at the time of s = 63. Then, the value of the average effectiveness fell to a minimum at s = 130. At this time, the turbulent spot transferred the heat from the hot plate only 5% above the laminar condition. However, it should be noted that the effect from the first hairpin vortex was completely eliminated. Thus, heat transfer through the turbulent spot, without other interference, can be measured after s > 130. Hence, this point is where the turbulent spot transitions in thermal behavior from the young state to the mature state. Beyond this point, the amount of heat transfer through the spot increases until it achieves the level of fully turbulent flow. 4.2. Structural analysis on the turbulent spot Fig. 12 shows the characteristics of the area of the turbulent spot body and the area of heat transfer during the dimensionless time from 45 to 182. Both areas are compared with the square of d⁄. Hence, the dimensionless area is obtained as A/d⁄2. The area, used previously in all thermal calculations, is the area of heat transfer. It is the whole area of the turbulent spot footprint, including the area of the spot body and the becalmed region and is proportional to the dimensionless time. Therefore, the size of the
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whole turbulent spot footprint increases when the turbulent spot propagates further downstream. Meanwhile, the area of the spot body was determined from the area counted from the leading edge to the trailing edge of the footprint. The location of trailing edge was identified as the mean point of all streamwise positions of the highest heat flux across the footprint from one side of the wingtips to the other. The results show that the area of the spot body increases with time and reaches a maximum of 670 at s 130. Then, the size of this area decreased again after s = 130 due to the change in the location of the trailing edge of the turbulent spot. During 0 6 s 6 130, the area of the spot body increased because the location of the trailing edge was held at the first two central streaks, while the leading edge traveled further downstream. At s 130, the position of the highest heat flux changed from the first two central streaks to the turbulent spot body. This change suddenly reduced the area of the spot body, as shown in Fig. 12 at s > 130. However, an increasing trend for this area was found at s > 180 because the new location of the trailing edge convected downstream with lower speed than the leading edge. This behavior also caused the elongation in the streamwise direction of the near wall turbulent spot body. Moreover, the difference between the heat transfer area and the spot body area represents the area of the calmed region, in which the thermal footprint returns from the highest turbulent state at the trailing edge to the laminar level of the surrounding area. Fig. 12 shows that the area of the calmed region also increases with time until s 130. This region then dramatically increases for the reasons previously
explained. This behavior can be further clarified by plotting the streamwise distance over time, as depicted in Fig. 13. Fig. 13 shows the plot of the streamwise distance and the dimensionless time of the centroid of the thermal area, the leading edge, the trailing edge, and the centroid of the spot body. Furthermore, the trailing edge data of the mature turbulent spot reported by Chaiworapuek et al. [17] is plotted as a dashed line during 150 < s < 190. From the plot, the streamwise distances of the centroid of the thermal area and the leading edge are linearly proportional to time. At this rate, the velocities at the leading edge and the centroid of the thermal area are (0.68 ± 0.03)Ua and (0.38 ± 0.02)Ua, respectively, where the uncertainties are obtained using the technique of Taylor [29]. Non-linear behavior is observed at both the centroid of the spot body and the trailing edge. However, the velocity of the spot body centroid can be determined as (0.65 ± 0.03)Ua when X > 40. Notice that in this period, the location of the highest heat flux or the trailing edge jumped from the first two central streaks to its body. Meanwhile, the trailing edge traveled downstream with the low speed of (0.19 ± 0.01)Ua, as illustrated by notation 1. It accelerated during 110 < s < 135 and propagated with the velocity of (1.06 ± 0.04)Ua during 135 < s < 160 at the notation 2. However, the velocity of the trailing edge became constant at (0.55 ± 0.02)Ua, in agreement with the velocity of the mature turbulent spot reported by Chaiworapuek et al. [17] in notation 3. Therefore, the trailing edge of the thermal footprint moved downstream non-linearly when the turbulent spot was in the young state, and its propagation behavior became linear when s > 160. The lateral spreading angle of the footprint can be determined by fitting the straight lines through the position of the spot wingtips, extracted from the contours of the heat flux. Fig. 14 shows the plot of the streamwise and spanwise distances of the upper and the lower wingtips. The spanwise growth of the turbulent spot footprint is approximately linear. The half spreading angle was (5.66 ± 0.02)°, which is in the range of the values obtained by other researchers [14–17,25]. It is noted that the manner of the spanwise growth is not related to the jump of the trailing edge location from the first two central streaks to the spot body. Furthermore, when the fitting line of both wingtips was traced back along the x-axis, it was found that the virtual origin was beyond the location of the spot generator, at X = 55.79. The growth rate of the turbulent spot can clearly be determined by the contributions of the spot velocities, the area, and the half spreading angle as a non-dimensional spot propagation parameter, r. Following the original concept of Emmons (1951), it is evaluated as follows:
Fig. 13. X–s plot of the centroid of the thermal area, the leading edge, the trailing edge, and the centroid of the spot body.
Fig. 14. Plot of the streamwise and spanwise distances of the upper and the lower wingtips.
Fig. 12. Comparison between the area of the spot body and the area of heat transfer.
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r¼
ðtan aÞ2 As CmW 2
ð15Þ
From the results, it was found that the non-dimensional spot propagation parameter was not constant during the early state of the turbulent spot, as shown in Fig. 15. It was approximately 0.06 at s = 45 and increased to a maximum at s 125. The main cause is that the location of the trailing edge slowly moves forward while the leading edge propagates downstream with the highest relative velocity. Hence, an increasing trend of r was found during s = 45–125. The maximum r was found at 0.1. This point shows the maximum growth of the footprint of the turbulent spot. However, the turning point was also found here because of the change in the location of the trailing edge. Thus, the non-dimensional spot propagation parameter later fell to 0.04 and seemed to remain constant at this value at s P 160. Connections are found between the thermal and structural analysis of the footprint of the turbulent spot. The results clearly show that, when the young turbulent spot body has the maximum growth, it transfers the minimum heat from the isothermal plate to the water, as a consequence of the effect of the spot initiation. Therefore, to study the characteristics of the mature turbulent spot, experiments should be conducted after s = 160. To check the reliability of the measurement technique employed in this study, the experiment was continued at the streamwise distance between 68 and 170 to observe the characteristics of the mature turbulent spot, and the results were compared with the findings of other researchers using different techniques [2,30–33]. The results showed that the footprint structure continues growing in both the streamwise and spanwise directions as depicted in Fig. 16. At the front edge of the mature turbulent spot, the middle frontal streak is piercing forward, and the structure forms an arrowhead-like shape, as observed by Zhong et al. [28]. The mature turbulent spot parameters, consisting of the velocities at the leading edge and trailing edge, the half spreading angle and the non-dimensional spot propagation parameter, were evaluated during s from 214.5 to 257.4 and compared with the values obtained by other researchers using different techniques [2,30– 33], as shown in Table 1. The table shows that the velocity at the leading edge of the mature turbulent spot in this study is slightly less in the others. The velocity at the trailing edge and the nondimensional spot propagation parameter of the mature spot are constant and found at 0.55 ± 0.02 and 0.088 ± 0.007, respectively. The half spreading angle changes from 5.66 ± 0.02° to 10 ± 0.02° when the spot becomes mature. This observation is consistent with the report by Kittichaikarn [34]. The table shows that all turbulent spot parameters are in agreement with other findings. Therefore,
Fig. 16. The contour of local spot effectiveness of the mature turbulent spot at s = 236 after the turbulent spot initiation.
Table 1 Parameters of mature turbulent spot obtained from the current study and other published research. Data source
CLE
CTE
a (°)
r
Schubauer and Klebanoff [2] Wygnanski et al. [30] Sankaran et al. [31] Gutmark and Blackwelder [32] Chong and Zhong [33] Current study
0.88
0.5
10
0.152
0.89 0.74 0.88
0.57 0.53 0.58
9.2 9 9
0.102 0.084 0.093
0.83 0.79 ± 0.03
0.54 0.55 ± 0.02
8.6 10 ± 0.02
– 0.088 ± 0.007
the results from the measurement technique using liquid crystals are reliable for this experiment on a turbulent spot.
5. Conclusions This study investigated the thermal and structural characteristics of the young turbulent spot footprint using thermochromic liquid crystals at a Reynolds number of approximately 75,000 in the recorded area. The turbulent intensity of the flow was 0.93%. The results showed that the thermal and structural dynamics of the young spot footprint were greatly influenced by the process of spot initiation. When the first hairpin vortex appeared after the spot injection, cold fresh water initially upstream of the spot generator directly impacted the heated surface. Through this process, the spot footprint provided high heat transfer, up to 10% above the laminar state. A hand-like shape was found instead of an arrowhead-like structure. When the influence from the first hairpin vortex decreased, the minimum spot heat transfer was found to be 5% above the laminar state. After this point, the heat transfer increased again when the spot moved further downstream. The velocities of the leading edge, thermal area, and spot body were found to be (0.68 ± 0.03)Ua, (0.38 ± 0.02)Ua, and (0.65 ± 0.03)Ua, respectively. The area of the spot body was in agreement with the non-dimensional spot propagation parameter, which approached the maximum at the time of the minimum heat transfer. This behavior was caused by the change in location of the spot’s trailing edge from the first two streaks, occurring early after the spot initiation to develop the turbulent spot body. When the effect of the spot initiation terminates, the thermal and structural characteristics of the turbulent spot enter a mature state at s = 160. After this time, the linear characteristics of the turbulent spot are obtained.
Conflict of interest Fig. 15. The non-dimensional spot propagation parameter through dimensionless time from 45 to 170.
We have no conflict of interest with other people and organizations.
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Acknowledgment The authors gratefully acknowledge the financial support from the Kasetsart University Research and Development Institute, Thailand.
[17]
[18]
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
On the thermal and structural characteristics of an artificially generated young turbulent spot Weerachai Chaiworapuek ⇑, Chawalit Kittichaikarn Department of Mechanical Engineering, Faculty of Engineering, Kasetsart University, Bangkok 10900, Thailand
a r t i c l e
i n f o
Article history: Received 8 February 2015 Received in revised form 20 August 2015 Accepted 17 September 2015 Available online 1 October 2015 Keywords: Turbulent spot Boundary layer transition Liquid crystals Heat transfer
a b s t r a c t The unsteady heat transfer from an isothermal wall to a water flow via a turbulent spot at an early stage was investigated experimentally using thermochromic liquid crystals. With a Reynolds number of approximately 75,000 at the test area, an artificially generated turbulent spot was induced by a water injection in a low-turbulence water tunnel with a turbulent intensity of 0.93%. The ratio of the spot heat transfer rate to the heat transfer rate existing without the spot was determined to evaluate the turbulent spot effectiveness. Selected images of both the Nusselt number and the spot effectiveness were illustrated to visualize the spot development. Interesting results were obtained soon after the spot initiation, as the spot provided an averaged heat transfer up to 10% above the laminar state. The results also indicated a significant influence of the water injection consequent to the non-linear behavior during the young state of the turbulent spot. The turning point from young to mature state was reported. Moreover, the spot footprint appeared as a hand-like shape and propagated downstream with 38% of the free stream velocity. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Turbulent spots occur under boundary layer transitions from laminar to turbulent flow and are responsible for the change in boundary layer properties and profile observed between the laminar and the turbulent state. The turbulent spot was first discovered by Emmons [1], who noticed a small turbulent patch on a water table. Emmons also showed that the turbulent spot could be generated directly by releasing a water droplet to strike the target surface. Later, Schubauer and Klebanoff [2] undertook experiments using a hot wire anemometer and found that the turbulent spot had an arrowhead shape, which propagated downstream with 88% and 50% of the free stream velocity at its leading and trailing edges, respectively. These researchers also found a stable state region similar to the laminar flow following the passage of the turbulent spot. This area was highly stable, and no breaking down was likely to appear. This region was ‘‘the recovery trail” or ‘‘the calmed region”. The results from many contributions [3–9] have also confirmed that the turbulent spot body has an arrowhead-like structure. Using a hot wire anemometer, Wygnanski et al. [4] suggested that the entire turbulent spot, based on the ensemble averaged analysis, ⇑ Corresponding author. E-mail address: [email protected] (W. Chaiworapuek). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.09.053 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
was a single large vortex. However, multiple sub-structures have been detected by different measurement techniques [5,8–13]. The growth mechanism of the turbulent spot in the streamwise and heightwise directions is due to the gulping entrainment process at its front interface and the nibbling process at its trailing edge [7]. Meanwhile, destabilization is considered the primary cause of the lateral growth of the turbulent spot [7,12]. In the bypass transition process, the turbulent spot can be directly initiated by injecting fluid into the flow or increasing the level of the free stream turbulence. Haidari and Smith [12] suggested that the initial hairpin vortex generated by the injection process was the primary vortex. This vortex then induces the subsidiary vortex and the secondary vortices, which were also considered main features of the turbulent spot. However, the spot would take some time to develop to a linear or mature state. This time becomes shorter with increasing Reynolds number [4]. The thermal structure of the turbulent spot was first investigated using a cold wire by Van Atta and Helland [18]. When it was compared with the contour of velocity perturbation, yielded by Zilberman et al. [19], the temperature and velocity fields were found to be strongly anti-correlated. Van Atta and Helland [18], Antonia et al. [20], and Chong and Zhong [21] were all in agreement that the near wall temperature inside the turbulent spot was below the local temperature before the spot arrival. This finding is consistent with the results reported by Zhong et al. [22],
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Nomenclature As CLE Cm CTE Ein Eg Est H hx k Nux Nu Pr Q Qns Qs q q qns qs Rex T t Tal
turbulent spot area (m2) turbulent spot velocity at leading edge propagation rate of the turbulent spot centroid turbulent spot velocity at trailing edge energy inflow (W) energy source (W) energy storage (W) Hue signal value heat transfer coefficient (W/m2 °C) thermal conductivity (W/m °C) local Nusselts number average Nusselts number Prandtl number rate of heat transfer (W) heat transfer rate of heated plate through the area of turbulent spot (W) rate of heat transfer through turbulent spot (W) heat flux (W/m2) average heat flux (W/m2) heat flux of heated plate through the area of turbulent spot (W) heat flux through turbulent spot (W) local Reynolds number temperature (°C) time, counted after spot initiation (s) temperature on aluminum plate (°C)
Sabatino and Smith [23], Chaiworapuek et al. [24], who employed the technique of temperature mapping using thermochromic liquid crystals. Typically, the shape of the turbulent spot’s footprint can be directly examined via the region of heat transfer [22]. Its foremost edge is defined as the leading edge, as shown in Fig. 1. The trailing edge and the half spreading angle can also be obtained as shown in the figure. The footprint underneath the turbulent spot propagates downstream with velocities in the range of 74–83% and 53–57% of the free stream velocity at the leading and trailing edges, respectively [16,17,25]. Additionally, the spreading half angle of the spot footprint was reported to be between 4° and 6.8° [14–17,25]. Chaiworapuek et al. [17] found that the arrowhead-like shape was first found when the leading edge reached the streamwise distance of x/d⁄ = 94.54 from the location of the spot generator. Before that, a hand-like structure was observed [17]. Sabatino and Smith [23] found that a peak in heat transfer at approximately 15% above the laminar condition occurred in the calmed region. This peak is caused by the cooler fluid entering from the upstream higher level, following Johnson [26].
Flow Direction
Wing tip
Half spreading angle Locus of wing tips
Virtual origin
Ta U Ua W X x x0 Dx Y y Dy Dz
free stream temperature (°C) velocity (m/s) free stream velocity (m/s) turbulent spot half width (m) dimensionless streamwise distance streamwise distance (m) streamwise location of the spot generator (m) streamwise length of each pixel (m) dimensionless spanwise distance spanwise distance (m) spanwise length of each pixel (m) heightwise length of each pixel (m)
Greek symbols a half spreading angle (°) dL laminar boundary layer thickness (m) d boundary layer displacement thickness at the spot generator (m) s dimensionless time es turbulent spot effectiveness es average turbulent spot effectiveness n unheated starting length (m) r non-dimensional spot propagation parameter
Free stream flow
Spot generator
PVC sheet
Bleeding channel Strip heaters Thermal insulator Supporter Pressure system
Steel plate
Fig. 2. Schematic of the test section in the low free stream turbulence water tunnel.
Hence, this study experimentally investigates the thermal and structural characteristics of a turbulent spot footprint. The test was conducted in a low free stream turbulence water tunnel with a Reynolds number of approximately 75,000 at the test area. To obtain all spot parameters, including the Nusselt number, the heat flux, the spot effectiveness, the velocities, the half spreading angle, and the non-dimensional spot propagation parameter, the thermochromic liquid crystals are used together with an analytical solution derived from the energy balance, following Chaiworapuek et al. [24]. All presented results provide valuable insight into the growth and heat transfer process of the turbulent spot footprint under the bypass transition.
Leading edge
Calmed region Trailing edge
Fig. 1. Schematic of a turbulent spot on a flat plate.
2. Experimental setup Fig. 2 shows a schematic diagram of the test section of the closed-loop low free stream turbulence water tunnel used in this
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study. The test section of 0.15 0.2 m was made from 10 mm thickness perspex to gain clear visibility throughout the experiment. The turbulent spot was directly initiated by injecting water through a 1 mm diameter hole, connecting with the pressure system at the distance of x0 = 0.3 m or x/d⁄ = 94.54 from the leading edge of the flat plate. At this point, the boundary layer displacement thickness d⁄ was measured as 3.17 mm using a Dantec fiber film probe type 55R11. A bleeding channel was located at the most upstream point of the section. The bleeding rate was adjustable by the gate valve, installed at an exit to provide a suitable condition for a new boundary layer, refreshed at the leading edge. The flat plate was a 3 mm thick aluminum plate for the low weight and good thermal conductivity. The nine strip heaters were set up below the aluminum plate to generate the isothermal condition of the flat plate. They were controlled using proportional-inte gral-derivative (PID) controllers that received local input signals from three stations of type-K thin leaf thermocouples. The thermocouples were finely glued on the top surface at 40, 58, and 78 cm from the leading edge. To prevent unwanted thermal loss, a high thermal resistance plate was laid beneath each heater. All strip heaters and the insulator were supported by a supporter and pressurized by a 3 mm thick steel plate at the lowest layer. On the top layer, a black polyvinyl chloride sheet with a thickness of 120 lm was well bonded with the aluminum plate. The plastic sheet had a thermal conductivity much lower than the aluminum plate; consequently, the heat transfer from the heaters was retarded and damped by the polyvinyl chloride sheet when conducted to an adjacent interface. Using the dimmer connected to each heater, power control of all heaters was performed independently, and a continuously isothermal condition of the flat plate was obtained. Temperature uniformity of the heated surface was confirmed by the uniform color of the liquid crystals coating the whole plate. Constant pressure of the spot generator was provided by the pressure system as shown in Fig. 3. A water tank was used to reserve the water for the injecting system. To control the pressure level of the injector, an air pump must fill a diaphragm tank with pressured air to obtain a suitable pressure level for the main system. The water injected from the spot generator was directly released through a solenoid valve with the duration of 0.02 s, activated by a programmable logic controller (PLC) board. Thus, this pressure level of the injected water was held constant through the experiment. In this study, the thermal footprint of the turbulent spot was qualitatively and quantitatively revealed by thermochromic liquid crystals coated at a thickness of approximately 20 lm on top of the plastic sheet. The liquid crystals used in the experiment were micro encapsulated Chiral Nematic crystals with an active range from 27 °C to 29 °C. They were mixed with CC300 binder to increase the water resistance. After coating, a continuous thin layer of clear vanish was applied to resist the degradation of the
Spot generator
P
reflective property of the liquid crystals by the water. Two 70 watt fluorescent bulbs, each with a diameter of 2.5 cm, were assembled beside the test section. Their white light, containing the full spectrum of wavelengths from red to blue, was used to reflect the colors from the crystals. Additionally, glossy reflectors were installed with each bulb to boost the light intensity. To acquire the image of the turbulent spot footprint, a video camera with a frame rate of 24 frames per second was installed 1.5 m above the test plate. At this height, the test area image of 0.31 m 0.12 m was obtained with a resolution of 281 pixels 681 pixels. The uniform flow was continuously supplied by a 2.5 HP centrifugal pump at the free stream velocity of Ua = 0.17 m s1, measured using a Nixon Streamflo velocity meter type 403 at the streamwise distance of x/d⁄ = 158 from the leading edge. This velocity corresponded to a Reynolds number of approximately 75,000 at the recorded area. At this position, a turbulence intensity of 0.93% was measured using a Dantec fiber film probe type 55R11. The velocity profile of the unheated laminar boundary layer was also obtained, as depicted in Fig. 4. All terms in the figure are defined in the Nomenclature section. 3. Mathematical analysis In the test section, the liquid crystals were fully coated on the heated wall for temperature measurement. The relationship between the scattered color and the temperature of the liquid crystals was established by the calibration process [17]. It was fit by the 3rd order polynomial equation as follows:
T ¼ 58:483H3 61:246H2 þ 23:156H þ 23:898
ð1Þ
It is noted that the scattered color is represented as H, which is the Hue signal from each pixel of the image. This equation gave a goodness of fit (R2) of 0.99873. In this study, the contour of the averaged temperature at each time step was used to construct the contours of the averaged heat transfer coefficient and averaged heat flux. It was calculated by pixel-by-pixel averaging of 30 individual turbulent spot footprints. Then, the heat transfer coefficient was determined from this averaged temperature contours using the energy balance method [24]. Initially, it was governed by the energy conservation equation as follows:
Ein þ Eg ¼ Est
ð2Þ
However, this method was used under the assumption that the temperature between the plastic sheet and the aluminum plate remains constant as the spot passes. Hence, with changing temperature, it was determined as follows:
Air pump
Diaphragm tank Air Water tank Water
Solenoid valve
P
Fig. 3. Schematic of the pressure system.
Fig. 4. Velocity profile, measured at the distance of x/d⁄ = 158 from the leading edge of the test plate.
W. Chaiworapuek, C. Kittichaikarn / International Journal of Heat and Mass Transfer 92 (2016) 850–858
T al ¼
Q Dz þT kðDxDyÞ
853
ð3Þ
Using this assumption, Q was evaluated based on the heat rate from the flat plate under the laminar flow, defined as follows:
Q ¼ hx ðDxDyÞðT T a Þ
ð4Þ
where hx is the local heat transfer coefficient on the flat plate with the unheated starting condition under laminar flow. Following Kays and Crawford [27], hx is given by
hx ¼
1=3 0:332Re1=2 k x Pr 3=4 1=3 x ½1 ðn=xÞ
ð5Þ
Fig. 6. The contour of the RGB signal at s = 0, 22.3, 33.5 and 44.6 after turbulent spot initiation.
Then, the local heat flux was simply calculated as
q ¼ hx ðT s T a Þ
ð6Þ
where q is the local heat flux from the flat plate to the water flow. Also, the local Nusselt number is defined as
hx x Nux ¼ k
ð7Þ
The local Nusselt number and heat flux obtained from the energy method are validated by comparison to the values determined from the theoretical formula (Eqs. (5) and (6)) as shown in Fig. 5. On the x-axis, the streamwise distance is presented in dimensionless form by comparison to the boundary layer displacement thickness at the spot generator as follows:
X ¼ ðx x0 Þ=d
ð8Þ
Both the Nusselt number and the heat flux are plotted through the distance, X, from 8 to 100, which is the area where the young turbulent spot occurs. It is found that the local Nusselt number and heat flux obtained from this study are consistent with the values achieved from theoretical prediction, which confirms the accuracy of the energy balance method. Moreover, the turbulent spot effectiveness, es, was determined from the ratio of the turbulent spot heat transfer rate, Qs, to the heat transfer rate existing without the spot, Qns. It was expressed as
es ¼
Qs q ¼ s Q ns qns
ð9Þ
Notice that this equation compares Qs and Qns through the area of the footprint. Thus, the rate of heat transfer in this study is alternately represented by the heat flux.
Fig. 5. The local Nusselt number and heat flux from the theoretical prediction and the experiment.
4. Results and discussion 4.1. Thermal analysis of the turbulent spot Fig. 6 presents the growth of the young turbulent spot after initiation beneath the main stream, which flows from left to right. The obtained colors were scattered by the liquid crystals. The streamwise distance on the x-axis starts from zero at the location of the spot generator and it is directly non-dimensionalized, as expressed previously. A spanwise distance on the y-axis was presented in dimensionless form as
Y ¼ y=d
ð10Þ
The zero distance is also set at the location of the spot generator. The value s is the dimensionless time, counted from the injection time. It is defined as the time in which the free stream water can move further with a streamwise distance of one boundary layer displacement thickness at the location of the spot generator, or
s¼
t d =U a
ð11Þ
The images in Fig. 6 were captured at s = 0, 22.3, 33.5, and 44.6 after spot initiation. The location of the spot generator was at the co-ordinate of X = 0 and Y = 0. At s = 0, the temperature of the test plate was kept at 28.3 °C shown in blue on the right side of all images in Fig. 6. The non-blue color, appearing from X = 0 to 8, showed the unheated starting zone. These images show the structure of the young turbulent spot after initiation by the spot generator. The arrival of the turbulent spot resulted in a color change from blue, as shown in the images at s = 0, 22.3, 33.5, and 44.6. It was found that the turbulent spot was first characterized by a 4-streak structure. The width of each streak was approximately two d⁄. Its two central streaks have very high heat transfer ability when the local Nusselt number is considered, as shown in Fig. 7. The streamwise and spanwise distances were newly established as X from 8 to 103 and Y from 18 to 18, respectively. This contour presents the local Nusselt number, evaluated by the color bar below at s from 44.6 to 200.9 with an interval of approximately 22. The blue shade appearing at X coordinates of 30 and 90 represents the location of the thin leaf thermocouples. Fig. 7 shows that the turbulent spot provides higher heat transfer ability than the unperturbed area beneath the laminar boundary layer. The area with the maximum Nusselt number first appeared as the two central streaks, quickly formed and propagating within the streamwise distance only X = 0–20. Immediately after the water injection, the first hairpin vortex formed from the injection process. This vortex brought cold water that was initially upstream of the spot generator to impact the heated surface following the flow direction of the vortex. The difference in temperature between the heated plate and the cold water was the cause of
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Fig. 7. The contour of the local Nusselt number at s = 44.6, 67, 89.3, 111.6, 133.9, 156.3, 178.6, and 200.9 after turbulent spot initiation.
the high-Nu region. However, the value of the maximum Nu decreased at s > 67. After the creation of the first or primary hairpin vortex, the subsidiary and secondary vortices were induced. The spot body was found to pull in all the surrounding water via the entrainment process, including the near wall hot water that initially relied on the heated surface. The colder water on the above layer was rapidly drawn towards the surface to replace the fluid that was now swept into the spot body. Hence, the process of heat transfer through the turbulent spot occurred. However, the influence of the temperature difference vanished completely by s > 200.9. At this time, it was found that the region with the maximum Nusselt number was newly located near the leading edge, inside the turbulent spot body. Notice that the maximum Nu at this point was approximately 360, which was lower than the maximum Nu occurring due to the first hairpin vortex. The maximum Nusselt number detected inside the footprint area is shown along the dimensionless time after spot initiation Fig. 8. The maximum Nusselt number increased to a maximum of 480 at s = 55 and was clearly located at the core of the two central streaks, as shown in Fig. 7. Then, the maximum Nusselt number decreased to 360 at s = 187. Furthermore, the average Nusselt number, Nu, through the spot footprint area, As, was also presented in each time step. It was evaluated by
Fig. 8. The maximum and average Nusselt numbers of the turbulent spot footprint.
Nus ¼
1 As
Z Nu dAs
ð12Þ
As the length of each image pixel was known, As in this study was determined by counting the pixel inside the turbulent spot footprint. The true shape of the footprint can be represented by the heat flux contour [16]. The boundary of the footprint was identified with the threshold of 10% of the maximum heat flux [28]. From the results, it was found that the average Nusselt number was highest at approximately 350 at s = 44.5. It then decreased to 270 and remained constant until s = 175. The structural behavior is further clarified by displaying the heat flux from the footprint. The contour of the local effectiveness of the turbulent spot, derived from the heat flux using Eq. (9), is depicted in Fig. 9. The main behavior of the thermal footprint from this contour is consistent with the behavior shown by the Nusselt number. The figure shows that the turbulent spot footprint first appeared as 4 streaks or a hand-like structure. It elongated in the streamwise direction at a higher rate than in the spanwise direction. The two central streaks transferred heat flux up to 25% above the unperturbed surface. Meanwhile, the local spot effectiveness at the two outer streaks was found to be approximately 20%. It is noted that the maximum effectiveness region occurred inside the two central streaks propagated within the distance and time of X = 20 and s = 89.3, respectively. Beyond this time, the magnitude of the heat transfer in this region decreased, and it later appeared as a part of the becalmed region located at the rear section of the footprint. Thus, the location of the maximum effectiveness jumps from the distance of X = 20 to 70 at the time of s = 111.6 to 133.9. This discontinuity shows that the effect of the first hairpin vortex, caused by the spot initiation, remained until s = 133.9. After this time, the region of maximum effectiveness appeared relatively near the leading edge of the footprint as it moved downstream with the turbulent spot. The maximum effectiveness was found at 25% above the laminar state again when the leading edge of the footprint reached X = 110. At s P 200.9, the value of the maximum effectiveness was still increasing as the spot convected further downstream. Fig. 10 shows the maximum and average heat fluxes of the turbulent spot footprint along the dimensionless time from 45 to 185 after the spot initiation. The maximum heat flux occurring inside
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855
Fig. 9. The contour of the local spot effectiveness at s = 44.6, 67, 89.3, 111.6, 133.9, 156.3, 178.6, and 200.9 after turbulent spot initiation.
Fig. 10. The maximum and average heat flux through the turbulent spot footprint.
the turbulent spot reached a peak of 690 W/m2 at s = 60 and decreased to a minimum of 220 W/m2 at s = 135. Unlike the Nusselt number, the maximum heat flux apparently increased again after s = 135. This result is in agreement with the average heat flux, q, which is determined from the relation
qs ¼
1 As
Z
q dAs
ð13Þ
The solid line that shows the average heat flux through the turbulent spot shows the same behavior as the maximum heat flux in Fig. 10. The peak was found to be approximately 240 W/m2 at s = 63. The minimum of 93 W/m2 was found at the same time with the dashed line and then slowly increased to 103 W/m2 at s = 185. To compare the amount of heat transfer through the turbulent spot with the heated surface under the laminar thermal boundary layer at the same spot area, the average spot effectiveness was proposed, as shown in Fig. 11. This parameter was evaluated as follows:
es ¼
1 As
Z
e dAs
ð14Þ
Fig. 11 shows that the characteristics of the average effectiveness are similar to the measured heat flux. After the spot initiation at s = 0, the perturbed area transferred heat from the heated sur-
Fig. 11. The average spot effectiveness through the dimensionless time from 45 to 185.
face to the water up to 10% above the laminar state at the time of s = 63. Then, the value of the average effectiveness fell to a minimum at s = 130. At this time, the turbulent spot transferred the heat from the hot plate only 5% above the laminar condition. However, it should be noted that the effect from the first hairpin vortex was completely eliminated. Thus, heat transfer through the turbulent spot, without other interference, can be measured after s > 130. Hence, this point is where the turbulent spot transitions in thermal behavior from the young state to the mature state. Beyond this point, the amount of heat transfer through the spot increases until it achieves the level of fully turbulent flow. 4.2. Structural analysis on the turbulent spot Fig. 12 shows the characteristics of the area of the turbulent spot body and the area of heat transfer during the dimensionless time from 45 to 182. Both areas are compared with the square of d⁄. Hence, the dimensionless area is obtained as A/d⁄2. The area, used previously in all thermal calculations, is the area of heat transfer. It is the whole area of the turbulent spot footprint, including the area of the spot body and the becalmed region and is proportional to the dimensionless time. Therefore, the size of the
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whole turbulent spot footprint increases when the turbulent spot propagates further downstream. Meanwhile, the area of the spot body was determined from the area counted from the leading edge to the trailing edge of the footprint. The location of trailing edge was identified as the mean point of all streamwise positions of the highest heat flux across the footprint from one side of the wingtips to the other. The results show that the area of the spot body increases with time and reaches a maximum of 670 at s 130. Then, the size of this area decreased again after s = 130 due to the change in the location of the trailing edge of the turbulent spot. During 0 6 s 6 130, the area of the spot body increased because the location of the trailing edge was held at the first two central streaks, while the leading edge traveled further downstream. At s 130, the position of the highest heat flux changed from the first two central streaks to the turbulent spot body. This change suddenly reduced the area of the spot body, as shown in Fig. 12 at s > 130. However, an increasing trend for this area was found at s > 180 because the new location of the trailing edge convected downstream with lower speed than the leading edge. This behavior also caused the elongation in the streamwise direction of the near wall turbulent spot body. Moreover, the difference between the heat transfer area and the spot body area represents the area of the calmed region, in which the thermal footprint returns from the highest turbulent state at the trailing edge to the laminar level of the surrounding area. Fig. 12 shows that the area of the calmed region also increases with time until s 130. This region then dramatically increases for the reasons previously
explained. This behavior can be further clarified by plotting the streamwise distance over time, as depicted in Fig. 13. Fig. 13 shows the plot of the streamwise distance and the dimensionless time of the centroid of the thermal area, the leading edge, the trailing edge, and the centroid of the spot body. Furthermore, the trailing edge data of the mature turbulent spot reported by Chaiworapuek et al. [17] is plotted as a dashed line during 150 < s < 190. From the plot, the streamwise distances of the centroid of the thermal area and the leading edge are linearly proportional to time. At this rate, the velocities at the leading edge and the centroid of the thermal area are (0.68 ± 0.03)Ua and (0.38 ± 0.02)Ua, respectively, where the uncertainties are obtained using the technique of Taylor [29]. Non-linear behavior is observed at both the centroid of the spot body and the trailing edge. However, the velocity of the spot body centroid can be determined as (0.65 ± 0.03)Ua when X > 40. Notice that in this period, the location of the highest heat flux or the trailing edge jumped from the first two central streaks to its body. Meanwhile, the trailing edge traveled downstream with the low speed of (0.19 ± 0.01)Ua, as illustrated by notation 1. It accelerated during 110 < s < 135 and propagated with the velocity of (1.06 ± 0.04)Ua during 135 < s < 160 at the notation 2. However, the velocity of the trailing edge became constant at (0.55 ± 0.02)Ua, in agreement with the velocity of the mature turbulent spot reported by Chaiworapuek et al. [17] in notation 3. Therefore, the trailing edge of the thermal footprint moved downstream non-linearly when the turbulent spot was in the young state, and its propagation behavior became linear when s > 160. The lateral spreading angle of the footprint can be determined by fitting the straight lines through the position of the spot wingtips, extracted from the contours of the heat flux. Fig. 14 shows the plot of the streamwise and spanwise distances of the upper and the lower wingtips. The spanwise growth of the turbulent spot footprint is approximately linear. The half spreading angle was (5.66 ± 0.02)°, which is in the range of the values obtained by other researchers [14–17,25]. It is noted that the manner of the spanwise growth is not related to the jump of the trailing edge location from the first two central streaks to the spot body. Furthermore, when the fitting line of both wingtips was traced back along the x-axis, it was found that the virtual origin was beyond the location of the spot generator, at X = 55.79. The growth rate of the turbulent spot can clearly be determined by the contributions of the spot velocities, the area, and the half spreading angle as a non-dimensional spot propagation parameter, r. Following the original concept of Emmons (1951), it is evaluated as follows:
Fig. 13. X–s plot of the centroid of the thermal area, the leading edge, the trailing edge, and the centroid of the spot body.
Fig. 14. Plot of the streamwise and spanwise distances of the upper and the lower wingtips.
Fig. 12. Comparison between the area of the spot body and the area of heat transfer.
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r¼
ðtan aÞ2 As CmW 2
ð15Þ
From the results, it was found that the non-dimensional spot propagation parameter was not constant during the early state of the turbulent spot, as shown in Fig. 15. It was approximately 0.06 at s = 45 and increased to a maximum at s 125. The main cause is that the location of the trailing edge slowly moves forward while the leading edge propagates downstream with the highest relative velocity. Hence, an increasing trend of r was found during s = 45–125. The maximum r was found at 0.1. This point shows the maximum growth of the footprint of the turbulent spot. However, the turning point was also found here because of the change in the location of the trailing edge. Thus, the non-dimensional spot propagation parameter later fell to 0.04 and seemed to remain constant at this value at s P 160. Connections are found between the thermal and structural analysis of the footprint of the turbulent spot. The results clearly show that, when the young turbulent spot body has the maximum growth, it transfers the minimum heat from the isothermal plate to the water, as a consequence of the effect of the spot initiation. Therefore, to study the characteristics of the mature turbulent spot, experiments should be conducted after s = 160. To check the reliability of the measurement technique employed in this study, the experiment was continued at the streamwise distance between 68 and 170 to observe the characteristics of the mature turbulent spot, and the results were compared with the findings of other researchers using different techniques [2,30–33]. The results showed that the footprint structure continues growing in both the streamwise and spanwise directions as depicted in Fig. 16. At the front edge of the mature turbulent spot, the middle frontal streak is piercing forward, and the structure forms an arrowhead-like shape, as observed by Zhong et al. [28]. The mature turbulent spot parameters, consisting of the velocities at the leading edge and trailing edge, the half spreading angle and the non-dimensional spot propagation parameter, were evaluated during s from 214.5 to 257.4 and compared with the values obtained by other researchers using different techniques [2,30– 33], as shown in Table 1. The table shows that the velocity at the leading edge of the mature turbulent spot in this study is slightly less in the others. The velocity at the trailing edge and the nondimensional spot propagation parameter of the mature spot are constant and found at 0.55 ± 0.02 and 0.088 ± 0.007, respectively. The half spreading angle changes from 5.66 ± 0.02° to 10 ± 0.02° when the spot becomes mature. This observation is consistent with the report by Kittichaikarn [34]. The table shows that all turbulent spot parameters are in agreement with other findings. Therefore,
Fig. 16. The contour of local spot effectiveness of the mature turbulent spot at s = 236 after the turbulent spot initiation.
Table 1 Parameters of mature turbulent spot obtained from the current study and other published research. Data source
CLE
CTE
a (°)
r
Schubauer and Klebanoff [2] Wygnanski et al. [30] Sankaran et al. [31] Gutmark and Blackwelder [32] Chong and Zhong [33] Current study
0.88
0.5
10
0.152
0.89 0.74 0.88
0.57 0.53 0.58
9.2 9 9
0.102 0.084 0.093
0.83 0.79 ± 0.03
0.54 0.55 ± 0.02
8.6 10 ± 0.02
– 0.088 ± 0.007
the results from the measurement technique using liquid crystals are reliable for this experiment on a turbulent spot.
5. Conclusions This study investigated the thermal and structural characteristics of the young turbulent spot footprint using thermochromic liquid crystals at a Reynolds number of approximately 75,000 in the recorded area. The turbulent intensity of the flow was 0.93%. The results showed that the thermal and structural dynamics of the young spot footprint were greatly influenced by the process of spot initiation. When the first hairpin vortex appeared after the spot injection, cold fresh water initially upstream of the spot generator directly impacted the heated surface. Through this process, the spot footprint provided high heat transfer, up to 10% above the laminar state. A hand-like shape was found instead of an arrowhead-like structure. When the influence from the first hairpin vortex decreased, the minimum spot heat transfer was found to be 5% above the laminar state. After this point, the heat transfer increased again when the spot moved further downstream. The velocities of the leading edge, thermal area, and spot body were found to be (0.68 ± 0.03)Ua, (0.38 ± 0.02)Ua, and (0.65 ± 0.03)Ua, respectively. The area of the spot body was in agreement with the non-dimensional spot propagation parameter, which approached the maximum at the time of the minimum heat transfer. This behavior was caused by the change in location of the spot’s trailing edge from the first two streaks, occurring early after the spot initiation to develop the turbulent spot body. When the effect of the spot initiation terminates, the thermal and structural characteristics of the turbulent spot enter a mature state at s = 160. After this time, the linear characteristics of the turbulent spot are obtained.
Conflict of interest Fig. 15. The non-dimensional spot propagation parameter through dimensionless time from 45 to 170.
We have no conflict of interest with other people and organizations.
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Acknowledgment The authors gratefully acknowledge the financial support from the Kasetsart University Research and Development Institute, Thailand.
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