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TIV and PIV based natural convection study over a square flat plate under stable stratification
International Journal of Heat and Mass Transfer 140 (2019) 660–670
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
TIV and PIV based natural convection study over a square flat plate under stable stratification Yifan Fan a,b, Yuguo Li b,⇑, Qun Wang b, Shi Yin b a b
College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, China Department of Mechanical Engineering, The University of Hong Kong, PokFuLam Road, Hong Kong Special Administrative Region
a r t i c l e
i n f o
Article history: Received 4 January 2019 Accepted 9 June 2019 Available online 14 June 2019 Keywords: Natural convection Heat transfer Stable stratification Thermal image velocimetry Isolated square heat surface Urban heat island circulation
a b s t r a c t Natural convection over an isolated horizontal plate in a stably stratified background condition is fundamental and challenging in industrial, atmospheric, geographical, and oceanic environmental studies, which, however, is still lack of investigation. In this study, the water tank modelling experiment with two main techniques, Thermal Image Velocimetry (TIV) and Particle Image Velocimetry (PIV), was conducted to investigate large eddy structures, the mean velocity field and heat transfer over an idealized square heated surface under the stable stratification. Shirking, growing and twisting eddy structures were identified in this study. By measuring the velocity fields at different horizontal planes with varied heights, an asymmetric flow structure over a square urban area is suggested by the PIV method to be characterized as the main diagonal inflow at lower levels and as the side outflow (perpendicular to edges) at upper levels. The mean flow speeds measured by TIV and PIV were quantitatively compared and analyzed by using cross-correlation coefficients. The low-level (2 mm or less above) velocity field measured by TIV also well presents the similar characteristics of near-surface diagonal inflow observed in the PIV measurements. Consequently, the reliability of the TIV method in this study is well verified for resolving the near-surface convective flow fields. The stable stratification is found to significantly impair the heat transfer efficiency as results indicated that the heat transfer in a stable environment is significantly weaker than that in a neutral environment by 71%. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection is represented by the well-known relationship between the Nusselt number Nu and the Rayleigh number Ra, as shown in Eq. (1).
Nu ¼ CRan
ð1Þ
where C and n are the coefficient and exponent respectively, and their values depend on the experimental conditions (laminar or turbulent and neutral or stable) and setups of heat sources (confined or non-confined, upward or downward and isolated or uniformly distributed and etc.). Nu and Ra are defined in Eqs. (2) and (3) respectively.
Ra ¼ gbDTL3 =ðmaÞ
ð2Þ
Nu ¼ hL=k
ð3Þ
⇑ Corresponding author. E-mail addresses: [email protected] (Y. Fan), [email protected] (Y. Li). https://doi.org/10.1016/j.ijheatmasstransfer.2019.06.031 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
where g is the acceleration of gravity (m s2). b is the thermal expansion rate (K1). DT is the temperature difference between heat source surface and the ambient environment (K). L is the characteristic length scale (m). m and a are kinematic viscosity (m2 s1) and thermal diffusivity (m2 s1) of the fluid respectively. h is the convective heat transfer coefficient. k is the conductivity (W K1 m1) of the fluid. The natural convection over an isolated horizontal flat plate in a neutral background environment has been studied by many researchers [42–49], which suggested that the exponent n should be 1/3 (or 0.33) and C has an average value of 0.152 in turbulent flows. Natural convection in a stable environment is crucial for studies on atmospheric flows, urban climate, and energy storage [16]. Focusing on the urban environment, haze and extreme heat waves are exaggerated when there is weak or even no background wind [4,36], which do serious damage to the health of city dwellers [38]. Under this condition, buoyancy-driven convective flows become dominated in urban heat island circulation (UHIC) [21,28,9,10]. It is important to understand flow fields, convective plume structures, and heat transfer characteristics of the UHIC in
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a stable atmospheric condition for the studies on urban-scale pollutant and heat transfer. With no background flow and stable stratification, many existing studies have employed fundamental water tank modelling to investigate the urban-scale airflow environment, by applying proper scaling parameters and considering urban area as an isolated heated surface [26,6,5,12] with main focuses of the mean flow field over a circular urban area. However, to the best of our knowledge, in a stable environment, few studies investigated the near-surface convective flow structures and heat transfer characteristics over an isolated square heated surface. Infrared camera, as a nonintrusive measurement, has been widely used to investigate convective flow on the heated surfaces effectively. The infrared camera samples time-sequential infrared images to study turbulent eddy structures and velocity fields [25,23], measure building surfaces’ heat flux [22], calculate sensible heat flux [13,31] and radiative heat flux [29,2] over the ground in the atmospheric boundary layer, and investigate the mechanism of multiphase flows [18]. It also has great potential medical applications, including studies of blood flow [39] and brain activity [1]. Inagaki et al. [23] successfully applied thermal image velocimetry (TIV) to measure the airflow on a wall surface and over a playground with turf to study the urban thermal and wind environment on the building and street scales. Besides using TIV technique, Particle Image Velocimetry (PIV), which uses the timesequence particle images to calculate the flow velocities, is also an efficient technique to quantitatively resolve the convective flow dynamics and turbulent eddy structures [8,15,33,40,41]. Aiming at the mentioned research gaps, this study addressed the challenging problems for the natural convection from an isolated square heated surface in the stably stratified environment, with the focuses of near-surface turbulent structures and convective flow fields. This study is expected to not only fundamentally reveal the near-surface flow/heat transfer features over a square area in the stable environment but also help for better understanding the convective flow dynamics of UHIC from an ideal 2-D square city. We novelly employed a reduced-scale water tank model with a salt-water-simulated stable stratification for modelling the stable atmospheric stratification. Similarities between the reduced-scale water tank model and the prototype of the atmospheric scale flow were guaranteed with the verified criteria used in [26,6,12]. TIV and PIV techniques are applied to study the mean flow fields, turbulent flow structures, and heat transfer characteristics due to the square heated surface in such a stable environment. Simultaneously, correlation and similarity of the TIV and PIV measurements are analyzed and quantitatively compared. Details of the methods and experimental setups are introduced in Section 2. The results are presented and discussed in Section 3. Our concluding remarks are summarized in Section 4.
2. Methods A reduced-scale idealized city model was embedded in a 0.4 0.4 0.5 m water tank (Fig. 1a). A square polyimide film heater was attached to a thin copper plate (120-mm sides; 0.2 mm deep) to simulate an ideal square urban area. As is illustrated in Fig. 1b, the heater (0.1 mm deep) and copper plate (0.2 mm deep) were thin to allow the temperature fluctuation above the copper plate to penetrate them and to be captured by the infrared camera below. The walls of the water tank were made of glass to provide good access for the laser sheet for application of PIV. The bottom was an acrylic plate for purposes of heat insulation. An infrared camera (FLIR SC660, FLIR Systems Inc., Wilsonville, OR) was used to capture the heater’s surface temperature (Fig. 1b). The resolution of the infrared camera is 640 480 pixels. Movement of the eddy structures follows the mean buoyancy-driven
flow above the copper plate in the near-surface region. Therefore, cross-correlation of two consecutive temperature fields (i.e., via TIV) can provide information on the mean velocity field by tracing the movement of the temperature fluctuations. The velocity fields on horizontal planes at various levels above the urban area were measured with PIV. The PIV system (Dantec Dynamics A/S, Denmark) consists of a high-speed camera (Speed Sense M140 with a resolution of 2560 1600 pixels) and a 10-W 532-nm continuous green laser (RayPower). The seeding particles for PIV application were 20-mm polyamide particles. Both the infrared camera and high-speed camera were operated at 30 Hz. Infrared images (i.e., TIV) and particle images (i.e., PIV) were analyzed with the same software (DynamicStudio; Dantec Dynamics A/S). The adaptive PIV algorithm was chosen with a certain range of interrogation window sizes (from 16 16 to 64 64 pixels, with an adjusting grid step size of 8 8 pixels). Based on the uncertainty estimation method described by Prasad et al. [34], the error of the PIV measurements in our application is less than 6%. The mean velocity fields obtained by TIV and PIV were quantitatively compared by calculation of the cross-correlation coefficients of speed matrices. The mean speed data obtained by TIV and PIV on horizontal planes are stored in matrices MT and MP, respectively. The cross-correlation coefficients c are obtained by Eq. (4).
cmax ¼ max½cðx0 ; y0 Þ
8 9 > > > < P MP ðx; yÞ MPx0 ;y0 M T ðx x0 ; y y0 Þ M T > = x;y ¼ max rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P 2 2> > > M P ðx; yÞ M Px0 ;y0 M T ðx x0 ; y y0 Þ M T > : ; x;y
x;y
ð4Þ where MPx0 ;y0 is the mean of MP ðx; yÞ in the region under the MT, and MT is the mean of matrix MT. The maximum correlation coefficient cmax in cðx0 ; y0 Þ represents the best fit between MT and MP. The power supplied to the heater was 64 W, so the average heat flux H0 over the urban area (a square region with 12-cm sides) was 4444 W m2. The bottom surface temperature of the heater, which was around 40.2 °C during the experiments, was measured with the infrared camera. To avoid any influence on the capture of temperature fluctuation, no thermal couple was used to measure the temperature of the upper surface (the interface of the water and the copper plate). Instead, it was represented by the bottom surface temperature (T w ¼ 40:2 °C) because the heater and copper were thin (0.3 mm in total) and the temperature difference between those two surfaces was small. The ambient water temperature was T 1 ¼ 18:2 °C, which was obtained with a microscale conductivity and temperature instrument (MSCTI; PME, Inc., Vista, CA; the temperature sensor is an FP07 thermistor). The room air and surrounding environment surface temperature Ta was the same as the water temperature. A 10 10 cm observation window, as is shown in Fig. 1b, provides access for the infrared camera to the heater. Therefore, heat leaks through this window into the air via natural convection and longwave radiation. Because the bottom of the heater faces downward, the air is hotter near the heater and thus forms a stable layer. In this situation, the heat leakage due to natural convection is estimated by conductive heat transfer in the air. In this case, the heat leakage from conductive power is Pc ¼ ka ADT=Dz, where ka ¼ 0:026 W m1 K1 is the conductivity of air, and DT can be estimated as T w T a = 22 °C. Dz can be estimated as the thickness of the bottom, where a pool of stagnant air was held in that cavity just below the heater (Fig. 1b). P c can thus be calculated and has an order of 0.3 W. The heat leakage from radiative power can be calculated by Pr ¼ AerðT w T a Þ4 based on the Stefan-Boltzmann law, where A = 0.01 m2 is the area of the
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observing window,
e = 0.8 [32] is the emissivity of the polyimide
film heater, r ¼ 5:67 108 W m2 K4 is the Stefan-Boltzmann constant, and T w and T a are the surface temperature of the polyimide film and its surroundings, respectively, in Kelvins. The radiative power Pr was around 104 W. Therefore, P c and P r were significantly smaller than the power supplied to the heater and
can thus be neglected. The supplied power can be regarded as being totally released through the upper surface of the copper plate into the salt water. Stable stratification was achieved with salt water, using the method described by Fan et al. [10], to simulate the static stable background conditions in the atmospheric boundary layer. The
Fig. 1. Experimental setup. (a) Schematic of the water tank. (b) Detailed schematic of the water tank bottom in cross section. (c) Background density profile.
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Fig. 1 (continued)
density profiles were measured with the MSCTI. The measured background density profile is shown in Fig. 1c. Based on the criteria in Lu et al. [26] and Catalano et al. [5], the similarity is achieved in our experiments between the experiment setup and the urban heat island circulation (UHIC) over an idealized square urban area. The linear fit of the density data, shown in Fig. 1c, gives a density gradient dq=dz of 46.5 kg m4, with an R-square of 0.9987. The background buoyancy frequency is thus calculated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N ¼ g q dq=dz = 0.675 s , where g is the gravity acceleration, 0
and q0 is the reference density. The convective velocity scale is given as uD ¼ ½gbDH0 =ðq0 cP Þ1=3 [26,6,5], where g is the gravity acceleration, b is the thermal expansion rate of water, H0 is the surface heat flux in the urban area, and q0 and cP are the reference density and specific heat capacity of water, respectively. uD has an order of 6:87 103 m s1. The Froude number Fr ¼ uD =ðNDÞ = 0.0847, which satisfies the similarity criteria based on studies of Lu et al. [26] and Fan et al. [10]. Therefore, the results in this study could help to better understand the fundamental large-scale urban-heat-island circulation produced by an ideal square city.
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Two types of eddy structure behaviors were identified: translation (Fig. 2a) and deformation (Fig. 2b and c). The growing, shrinking, and twisting eddy structures can be observed in the deformation process shown in Fig. 2b and c. The twisting and translation of eddy structures are caused by the mean flow, which is maintained by the difference in pressure between the heated urban area and the surrounding rural area. The same phenomenon was observed by [40] and Fan et al. [11] as ‘‘bent over plumes” in the convective boundary layer from a vertical cross-section perspective. Shrinking and growing eddy structures indicate the surface renewal process [14,24,17], which is a widely studied phenomenon of heat and mass transfer at the interfaces, such as between seawater and air or between a solid surface and the adjacent fluid layer. Both hot and cold eddy structures present growing and shrinking characteristics. The growing hot eddy structures (Fig. 2b) indicate that the fluid is heated by the surface and that the thermal boundary is becoming thicker, forming a larger thermal plume. In contrast, the shrinking hot eddy structures are caused by the rising and detachment of hot fluid parcels from the surface, which results in the invasion of cold surrounding fluid into the original position of the thermal plumes. The impact of cold fluid parcels on the surface can lead to the growth of the blue eddy structures. The shrinking blue eddy structures are caused by heat from the hot surface, which results in the gradually decreasing sizes of cold fluid parcels. These identified characteristics of eddy structures can be applied to model sensible and latent heat flux and cloud formation over urban and rural areas. These behaviors of eddy structures are also important for applications on heat flux measurement using a surface renewal model. It has great potential on heat flux measurements over the earth’s surface to predict climate change and on the heat exchangers of refrigerators or boilers to improve the efficiency of energy use and to study the mechanism of heat transfer on surfaces. The bulk heat transfer coefficient over the heated surface in the current experimental setup is also obtained quantitatively based on the parameters given in Section 2. In our study, the Nusselt number under the stable stratification Nus can be calculated to be 40 according to Eq. (3) while the Rayleigh number Ras is given as 7:7 108 based on Eq. (2). With the same conditions but in the neutral environment, the Nusselt number Nun is obtained as 1=3 ). Therefore, the relative differ140 (Nun ¼ 0:152 7:7 108 ence of Nuesselt numbers under different conditions has an order of ðNun Nus Þ=Nun 100% ¼ 71% . It is shown that the heat transfer coefficient in a stable environment is significantly lower than that in a neutral environment by 71%.
3. Results and discussion 3.1. Eddy structures and heat transfer
3.2. Analysis of velocity fields obtained by PIV
Eddy structures over the heated urban area were visualized by mapping the temperature fluctuation field. The characteristics of those eddy structures are shown in Fig. 2. The frames in Fig. 2 were taken when the flow reached a quasisteady state. Fig. 2a(i) and b(i) are the same figure, but they were drawn separately to clearly demonstrate different types of eddies. The legends are fixed between 0.5 °C and 1.2 °C. Red1 and blue represent positive and negative temperature fluctuations, respectively. In the following text, we name the eddy structures shown by positive and negative temperature fluctuation as hot eddy structures and cold eddy structures, respectively. Fig. 2a(i) and (ii) show the temperature fluctuation fields after heating for 121 s and 123 s, respectively. The filter was a 30-s mean temperature field.
As is shown in Fig. 3, the mean velocity fields on horizontal planes were measured by PIV at various heights. The contour in Fig. 3 represents a 30-s mean speed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (s ¼ u2 þ v 2 ) on horizontal planes, where u and v are the horizontal velocity components in the x and y directions, respectively. Arrows represent velocity vectors. The horizontal and vertical axes are non-dimensional coordinates in the x and y directions respectively. D is the side length of the heated urban area. The geometric center of the urban area is set as the coordinate origin (0, 0). The acquisition frequency was 30 Hz and the duration 30 s for each data set. Fig. 3 shows the 30-s average results. It can be seen that the flow above the urban area is characterized as diagonal inflow at the lower levels (2, 4, and 6 mm) and as side outflow perpendicular to the edges at the upper level (12 mm), which supports the findings of Fan et al. [10,11]. The inflow region is thicker in the vertical direction along diagonals
1 For interpretation of color in Figs. 2 and 6, the reader is referred to the web version of this article.
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Fig. 2. (a) Translation, and (b and c) deformation, including growing, shrinking, and twisting, of eddy structures. Refer to the video in the supplementary material for a clearer demonstration of eddy dynamics.
and thus creates a ‘‘ditch-like region” between two diagonals. The strong diagonal inflows produce high-pressure ridges, and relatively low flow-resistance regions then form on four side regions,
resulting in outflow jets that resemble a ‘‘four-leaf clover.” It should be noted that the convergent center does not coincide with the geometric center of the heated urban area in this study (Fig. 3a)
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Fig. 3. Mean velocity fields over heated urban area measured by PIV on horizontal planes (a) 2 mm, (b) 4 mm, (c) 6 mm, and (d) 12 mm above the level of the heated surface. White dashed squares mark the urban area’s location.
for two possible reasons. One is that the thin copper plate (0.2 mm) is deformed under the pressure of salt water and thus forms a small incline. This inclination helps the convergent center to shift uphill akin to the UHIC on a slope [30]. The other reason is the heater’s lack of uniformity due to the manufacturing process. The characteristic of diagonal inflow is an advantage for comparison of the PIV and TIV results because it is easy to recognize. The maximum speed over the urban area has an order of 4 103 m s1. The mean velocity obtained by TIV represents the eddy movement near the surface. Therefore, it will be mainly compared with the PIV results at 2 mm (Fig. 3a) in Section 3.3. The standard deviations of the u and v components are plotted in Figs. 4 and 5, respectively. The contours in Figs. 4 and 5 represent the standard deviations of u or v velocity components. Velocity vectors (arrows) are also plotted to provide information on the velocity field at corresponding heights. The maximum standard deviation of the velocity components is around 1.2 103 m s1, which is of the same order as that of the mean speed (4 103 m s1). The standard deviations of the velocity components also present larger values along the diagonals at lower levels and in the side regions at upper levels. However, there is a different phenomenon than that of the mean speed field. As is shown in Figs. 4b, c and 5b, c, the standard deviation field of the velocity components also presents a ‘‘four-leaf
clover” shape, even those actually on lower levels in the inflow region (4- and 6-mm heights). However, the ‘‘four-leaf clover” shape only presents at the upper level (Fig. 3d; 12-mm height) for contours of the mean speed. This indicates that the outflow jets can influence the flow at 4- and 6-mm heights. High standard deviation values, which indicate a highly turbulent region, appear in the regions where the mean speed is high. It is interesting to observe that the standard deviation of the u velocity component is greater in the x direction than that in the y direction (Fig. 4d). Similarly, the standard deviation of the v velocity component demonstrates the same trend (Fig. 5d), which is attributed to the main flow direction (i.e., high fluctuation in the main flow direction). 3.3. Comparison of TIV and PIV data Fig. 6 shows the mean speed calculated by the TIV method, based on sequential infrared images. To compare the TIV data with those of PIV, the 1-s average velocity field obtained by PIV at the 2mm height is plotted in Fig. 7. As shown in Fig. 6, the characteristics of the main diagonal inflows can be successfully captured by TIV calculation. However, the TIV and PIV results have two main different aspects. One is that the short-term average velocity fields obtained by TIV (1-s, 10-s,
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Fig. 4. Standard deviations of the u velocity component at heights of (a) 2 mm, (b) 4 mm, (c) 6 mm, and (d) 12 mm, respectively.
and 30-s average fields shown in Fig. 6a, b, and c respectively) are less smooth than those obtained by PIV (Fig. 7). The diagonal inflow cannot be observed at all in the 1 s-average velocity field obtained by TIV (Fig. 6a). The other aspect is the magnitudes of the measured speed. The maximum speed over the urban area measured by TIV has an order of 4 105 m s1, which is smaller than that measured by PIV by two orders of magnitude. There are several reasons for the differences in the measurement principles for PIV and TIV. First, the temperature fluctuation captured by TIV does not necessarily represent the movements of the eddy structures at the 2-mm height, where the velocity field is measured by PIV. Those captured temperature fluctuations reflect movements of the eddy structures near the surface [19,20,23], but it is difficult to identify the exact height. According to Puthenveettil and Arakeri [35], the thermal boundary layer depth dt has an order of Rac , where c (0.28–0.35 in [27] 0.309 in [7] 0.29 in [3] 0.288 in [37] is a coefficient and Ra ¼ gbDTD3 =ðmaÞ is Rayleigh number. D = 12 cm is the side length of the urban area. g is the acceleration of gravity, and b is the thermal expansion rate of water. DT ¼ T w T 1 is the temperature difference between the heated surface and the ambient fluid. a is the thermal diffusivity of water, and m is the kinematic viscosity. Therefore, the Ra in our experiments had an order of 7:6 108 . The thermal boundary layer depth dt can thus be estimated based on the coefficients mentioned above and ranges from 0.78 to 3.26 mm. Because the copper plate’s
surface is a no-slip boundary, the velocity at the surface is zero. Therefore, movements of those visualized eddy structures represent the flow between 0 and 0.78 or 3.26 mm above the copper plate. Second, the movements of the large eddy structures are slower than the instantaneous background flow. Small eddy structures can follow the main flow well, but their temperature fluctuations are not sufficiently large to penetrate the heater and copper plate and be captured by the infrared camera. This means that small (i.e., not large enough to penetrate) turbulent thermal structures are difficult to measure with the TIV method and that the average time should be large to get a more representative mean flow field. An average time of 30 s is needed, as shown in Figs. 8 and 9; this will be analyzed later in detail. Although the heater and copper plate were as thin as possible, small eddy structures are still filtered. Large eddy structures are successfully measured. The dynamic interaction between large thermal plumes and the mean background flow was studied by Hunt [50] and by Fan et al. [11], and their studies suggested that plumes can be bent over by the mean background flow. Thermal plumes can also be advected by the mean flow, but the movement speed of the plume structures along the main flow direction is lower than that of the main flow. Third, the quantity of large eddy structures is not sufficient, leading to a lower time-averaged speed. If no movement of large eddy structures is captured by the infrared camera during a certain
Y. Fan et al. / International Journal of Heat and Mass Transfer 140 (2019) 660–670
Fig. 5. Standard deviations of the
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v velocity component at heights of (a) 2 mm, (b) 4 mm, (c) 6 mm, and (d) 12 mm, respectively.
time slot at a certain location, the mean flow calculated by TIV would be low or even zero at that specific time slot and location. Therefore, when the speed is averaged over time, the mean speed would be low due to those small values. This phenomenon is supported by the results that a larger average time leads to a lower mean speed, as shown in Fig. 6, in which the area with a higher speed (red regions) decreases as the average time increases (from Fig. 6a to d). Although the magnitude of the flow field obtained by TIV differs from that of PIV, the characteristics of the flow field are similar. For a quantitative comparison of the PIV and TIV mean speeds, crosscorrelation coefficients were calculated based on Eq. (4), which describes the similarity between two sets of data. The detailed parameters of PIV and TIV cases are summarized in Tables 1 and 2, respectively. The cross-correlation coefficients between different TIV datasets and different PIV datasets are shown in Fig. 8. In Fig. 8, the horizontal axis is the average time (range, 1–180 s) of the TIV datasets (TIV1-TIV12), and the vertical axis shows the cross-correlation coefficients of TIV with the corresponding PIV data sets. The data show that the cross-correlation coefficients increase quickly with the average time on the left hand of the vertical dashed line and then change slowly when the average time exceeds 30 s. This trend indicates that 30 s is a proper value to calculate the mean velocity field with the TIV method. The average
time of 30 s is also supported by scaling analysis. The time scale of UHIC in our experiments is t = D/s, where D = 120 mm is the side length of the urban area and s is the convective speed, which has an order of 4 103 m s1, shown in Fig. 3. Therefore, t = 120 103 m/4 103 m s1 = 30 s. Another phenomenon is that only the cross-correlation coefficients between the TIV data sets and the PIV2mm data sets are larger than 0.75 (above the horizontal dashed line). This result suggests that the mean velocity field calculated by TIV represents the mean flow at or below the 2-mm height. The results of PIV2mm1s and PIV2mm, shown in Fig. 8, are similar, indicating that a 1-s average in the PIV calculation can represent the mean flow well. This conclusion is also supported by the cross-correlation coefficients between different PIV data sets, as shown in Table 3. As shown in Table 3, the cross-correlation coefficient between PIV2mm1s and PIV2mm is 0.995, which is rather high and explains the similarity between PIV2mm and PIV2mm1s in Fig. 8. This further verifies that the 1-s average of the velocity field in the PIV calculation can represent the mean flow well in a water tank model of UHIC. The cross-correlation coefficient between PIV4mm and PIV2mm is also relatively high (0.957) because both are located in the inflow region of the UHIC at lower levels, and both present similar characteristics of main diagonal inflow. In contrast, the totally different flow types between PIV2mm (diagonal inflow) and PIV12mm (side outflow) lead to a low correlation coefficient
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Fig. 6. Mean speed calculated based on infrared images with averaging time of (a) 1 s, (b) 10 s, (c) 30 s, and (d) 180 s. White-filled arrows indicate the main flow directions.
Fig. 7. (a) 1-s average velocity field over heated urban area measured by PIV at the 2-mm height. (b) Enlargement of velocity field, marked by black dashed square in (a), to show the same scope of TIV view. White-dashed square indicates the location of the heated square area.
(0.67). The cross-correlation coefficients between PIV and TIV for different time slots are plotted in Fig. 9. As shown in Fig. 8, 30 s is a proper value to calculate the mean velocity in the TIV application. Therefore, consecutive data sets
with a 30-s interval were used in Fig. 9. No obvious increasing or decreasing trend for coefficients is seen in Fig. 9, which indicates that the flow reaches a quasi-steady state after 120 s of heating. The coefficients are high (above the horizontal dashed line) only
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Y. Fan et al. / International Journal of Heat and Mass Transfer 140 (2019) 660–670 Table 2 Parameters for TIV data (acquisition frequency, 30 Hz).
Fig. 8. Cross-correlation coefficients between different TIV datasets and different PIV data sets. Vertical and horizontal blue dashed lines mark the average time of 30 s (TIV7) and the cross-correlation coefficient of 0.75, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
TIV datasets
Time slots of datasets
Duration and average time (s)
TIV1 TIV2 TIV3 TIV4 TIV5 TIV6 TIV7 TIV8 TIV9 TIV10 TIV11 TIV12 TIV13 TIV14 TIV15 TIV16 TIV17
120–121 s 120–122 s 120–123 s 120–125 s 120–130 s 120–140 s 120–150 s 120–180 s 120–210 s 120–240 s 120–270 s 120–300 s 150–180 s 180–210 s 210–240 s 240–270 s 270–300 s
1 2 3 5 10 20 30 60 90 120 150 180 30 30 30 30 30
Table 3 Cross-correlation coefficients among various PIV data sets.
PIV2mm1s PIV2mm PIV4mm PIV6mm PIV12mm
PIV2mm1s
PIV2mm
PIV4mm
PIV6mm
PIV12mm
1 0.995 0.957 0.796 0.67
0.995 1 0.956 0.796 0.66
0.957 0.956 1 0.88 0.68
0.796 0.796 0.88 1 0.62
0.67 0.66 0.68 0.62 1
4. Conclusions
Fig. 9. Cross-correlation coefficients between 30-s mean speed calculated by PIV and that calculated by TIV.
Table 1 Parameters for PIV data (acquisition frequency, 30 Hz). Note that PIV2mm, PIV4mm, PIV 6 mm, and PIV 12 mm were not adopted in the same time slot because velocity fields at different levels cannot be obtained simultaneously and must be measured in sequence. Heating began at 0 s. PIV datasets
Time slots of data sets
Duration and average time (s)
Distance between measuring plane and surface (mm)
PIV2mm1s PIV2mm PIV4mm PIV6mm PIV12mm
120–121 s 120–150 s 225–255 s 332–362 s 442–472 s
1 30 30 30 30
2 2 4 6 12
for those with PIV measurements at the 2-mm height (PIV2mm and PIV2mm1s), which also suggests that the TIV measurements in our study are more representative for mean flow fields 2 mm or less above the surface.
In this study, TIV and PIV are applied in investigations of the natural convection from an isolated square heated surface in the stably stratified environment. Two different behaviors of eddy structures, including translation and deformation, were observed, along with three types of deformation processes: growing, shrinking, and twisting. These identified behaviors are important for heat and mass transfer studies in the boundary layer over surfaces and have great potential applications in the measurement of heat flux over the earth’s surface and in heat exchangers. The bulk heat transfer coefficient in a neutral environment is found to be higher by 250% than that in the studied stable environment. Besides, the mean flow field and turbulent characteristics on different horizontal planes were further measured by PIV. The results suggest that the asymmetric flow structure over a square urban area, which is characterized as the main diagonal inflow at lower levels and as the side outflow (perpendicular to edges) at upper levels. The TIV technique depends on the transient temperature variations transferred by the flow eddies. The limitation is that small turbulent eddy structures can hardly be captured by TIV because small temperature fluctuations are not significant enough to penetrate the heater and the copper plate. Therefore, a longer average time (30 s in this study) is required for TIV application to obtain a typical mean flow field. In comparison, a 1-s average is enough for PIV measurements because the PIV technique directed measure the seeding particle displacement with the flow motions. The mean speeds over the square heat surface from PIV and TIV are compared by using cross-correlation coefficients to quantify the similarity. The mean speed measured by TIV with an average time above 30 s has a high correlation (above 0.75) with that measured by PIV at 2 mm above the surface. The low-level (2 mm or less above) velocity field measured by TIV also well presents the similar characteristics of near-surface diagonal inflow observed in the PIV measurements. These similar results verify the capability of TIV for resolving the near-surface convective flow fields.
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Declaration of Competing Interest None. Acknowledgments This study is financially supported by an RGC grant (No. 17202618) and NSFC grant (No. 51778555). We are grateful for the technical support from Mr. Leung and Mr. Tam at HKU. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijheatmasstransfer.2019.06.031. References [1] N. Afrashteh, S. Inayat, M. Mohsenvand, M.H. Mohajerani, Optical-flow analysis toolbox for characterization of spatiotemporal dynamics in mesoscale optical imaging of brain activity, Neuroimage 153 (2017) 58–74. [2] T. Astarita, G.M. Carlomagno, Infrared Thermography for Thermo-fluiddynamics, Springer, Berlin, 2013, p. 236. [3] A. Belmonte, A. Tilgner, A. Libchaber, Temperature and velocity boundary layers in turbulent convection, Phys. Rev. E 50 (1994) 269–279. [4] C. Cao, X. Lee, S. Liu, N. Schultz, W. Xiao, M. Zhang, L. Zhao, Urban heat islands in China enhanced by haze pollution, Nat. Commun. 7 (2016) 12509. [5] F. Catalano, A. Cenedese, S. Falasca, M. Moroni, Numerical and experimental simulations of local winds, in: National Security and Human Health Implications of Climate Change, Springer, 2012, pp. 199–218, https://doi.org/ 10.1007/978-94-007-2430-3_17. [6] A. Cenedese, P. Monti, Interaction between an inland urban heat island and a sea-breeze flow: a laboratory study, J. Appl. Meteorol. 42 (2003) 1569–1583. [7] X. Chavanne, F. Chilla, B. Chabaud, B. Castaing, B. Hebral, Turbulent RayleighBénard convection in gaseous and liquid He, Phys. Fluids 13 (2001) 1300– 1320. [8] F. Corvaro, M. Paroncini, M. Sotte, Experimental PIV and interferometric analysis of natural convection in a square enclosure with partially active hot and cold walls, Int. J. Therm. Sci. 50 (9) (2011) 1629–1638, https://doi.org/ 10.1016/j.ijthermalsci.2011.03.029. [9] Y. Fan, Y. Li, J. Hang, K. Wang, X. Yang, Natural convection flows along a 16storey high-rise building, Build. Environ. 107 (2016) 215–225. [10] Y. Fan, Y. Li, S. Yin, Interaction of multiple urban heat island circulations under idealised settings, Build. Environ. 134 (2018) 10–20. [11] Y. Fan, J.C.R. Hunt, S. Yin, Y. Li, Mean shear flow in recirculating turbulent urban convection and the plume-puff eddy structure below stably stratified inversion layers, Theor. Appl. Climatol. (2018) 1–15. [12] Y. Fan, Y. Li, S. Yin, Non-uniform ground-level wind patterns in a heat dome over a uniformly heated non-circular city, Int. J. Heat Mass Transf. 124 (2018) 233–246. [13] A. Garai, J. Kleissl, Air and surface temperature coupling in the convective atmospheric boundary layer, J. Atmos. Sci. 68 (2011) 2945–2954. [14] C.S. Garbe, U. Schimpf, B. Jähne, A surface renewal model to analyze infrared image sequences of the ocean surface for the study of air-sea heat and gas exchange, J. Geophys. Res.-Oceans 109 (2004) C08S15. [15] S. Grafsrønningen, A. Jensen, B. Anders Pettersson Reif, PIV investigation of buoyant plume from natural convection heat transfer above a horizontal heated cylinder, Int. J. Heat Mass Transf. 54 (23–24) (2011) 4975–4987, https://doi.org/10.1016/j.ijheatmasstransfer.2011.07.011. [16] S.K. Gupta, Y. Jaluria, An experimental and analytical study of thermal stratification in an enclosed water region due to thermal energy discharge, Energy Convers. Manage. 22 (1982) 63–70. [17] E. Haghighi, D. Or, Turbulence-induced thermal signatures over evaporating bare soil surfaces, Geophys. Res. Lett. 42 (2015) 5325–5336. [18] G. Hetsroni, R. Rozenblit, Thermal patterns on a heated wall in vertical airwater flow, Int. J. Multiph. Flow 26 (2000) 147–167. [19] G. Hetsroni, R. Rozenblit, L. Yarin, A hot-foil infrared technique for studying the temperature field of a wall, Meas. Sci. Technol. 7 (1996) 1418–1427. [20] G. Hetsroni, J. Zakin, A. Mosyak, Low-speed streaks in drag-reduced turbulent flow, Phys. Fluids 9 (1997) 2397–2404. [21] J. Hidalgo, G. Pigeon, V. Masson, Urban-breeze circulation during the CAPITOUL experiment: observational data analysis approach, Meteorol. Atmos Phys. 102 (2008) 223–241.
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
TIV and PIV based natural convection study over a square flat plate under stable stratification Yifan Fan a,b, Yuguo Li b,⇑, Qun Wang b, Shi Yin b a b
College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, China Department of Mechanical Engineering, The University of Hong Kong, PokFuLam Road, Hong Kong Special Administrative Region
a r t i c l e
i n f o
Article history: Received 4 January 2019 Accepted 9 June 2019 Available online 14 June 2019 Keywords: Natural convection Heat transfer Stable stratification Thermal image velocimetry Isolated square heat surface Urban heat island circulation
a b s t r a c t Natural convection over an isolated horizontal plate in a stably stratified background condition is fundamental and challenging in industrial, atmospheric, geographical, and oceanic environmental studies, which, however, is still lack of investigation. In this study, the water tank modelling experiment with two main techniques, Thermal Image Velocimetry (TIV) and Particle Image Velocimetry (PIV), was conducted to investigate large eddy structures, the mean velocity field and heat transfer over an idealized square heated surface under the stable stratification. Shirking, growing and twisting eddy structures were identified in this study. By measuring the velocity fields at different horizontal planes with varied heights, an asymmetric flow structure over a square urban area is suggested by the PIV method to be characterized as the main diagonal inflow at lower levels and as the side outflow (perpendicular to edges) at upper levels. The mean flow speeds measured by TIV and PIV were quantitatively compared and analyzed by using cross-correlation coefficients. The low-level (2 mm or less above) velocity field measured by TIV also well presents the similar characteristics of near-surface diagonal inflow observed in the PIV measurements. Consequently, the reliability of the TIV method in this study is well verified for resolving the near-surface convective flow fields. The stable stratification is found to significantly impair the heat transfer efficiency as results indicated that the heat transfer in a stable environment is significantly weaker than that in a neutral environment by 71%. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection is represented by the well-known relationship between the Nusselt number Nu and the Rayleigh number Ra, as shown in Eq. (1).
Nu ¼ CRan
ð1Þ
where C and n are the coefficient and exponent respectively, and their values depend on the experimental conditions (laminar or turbulent and neutral or stable) and setups of heat sources (confined or non-confined, upward or downward and isolated or uniformly distributed and etc.). Nu and Ra are defined in Eqs. (2) and (3) respectively.
Ra ¼ gbDTL3 =ðmaÞ
ð2Þ
Nu ¼ hL=k
ð3Þ
⇑ Corresponding author. E-mail addresses: [email protected] (Y. Fan), [email protected] (Y. Li). https://doi.org/10.1016/j.ijheatmasstransfer.2019.06.031 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
where g is the acceleration of gravity (m s2). b is the thermal expansion rate (K1). DT is the temperature difference between heat source surface and the ambient environment (K). L is the characteristic length scale (m). m and a are kinematic viscosity (m2 s1) and thermal diffusivity (m2 s1) of the fluid respectively. h is the convective heat transfer coefficient. k is the conductivity (W K1 m1) of the fluid. The natural convection over an isolated horizontal flat plate in a neutral background environment has been studied by many researchers [42–49], which suggested that the exponent n should be 1/3 (or 0.33) and C has an average value of 0.152 in turbulent flows. Natural convection in a stable environment is crucial for studies on atmospheric flows, urban climate, and energy storage [16]. Focusing on the urban environment, haze and extreme heat waves are exaggerated when there is weak or even no background wind [4,36], which do serious damage to the health of city dwellers [38]. Under this condition, buoyancy-driven convective flows become dominated in urban heat island circulation (UHIC) [21,28,9,10]. It is important to understand flow fields, convective plume structures, and heat transfer characteristics of the UHIC in
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a stable atmospheric condition for the studies on urban-scale pollutant and heat transfer. With no background flow and stable stratification, many existing studies have employed fundamental water tank modelling to investigate the urban-scale airflow environment, by applying proper scaling parameters and considering urban area as an isolated heated surface [26,6,5,12] with main focuses of the mean flow field over a circular urban area. However, to the best of our knowledge, in a stable environment, few studies investigated the near-surface convective flow structures and heat transfer characteristics over an isolated square heated surface. Infrared camera, as a nonintrusive measurement, has been widely used to investigate convective flow on the heated surfaces effectively. The infrared camera samples time-sequential infrared images to study turbulent eddy structures and velocity fields [25,23], measure building surfaces’ heat flux [22], calculate sensible heat flux [13,31] and radiative heat flux [29,2] over the ground in the atmospheric boundary layer, and investigate the mechanism of multiphase flows [18]. It also has great potential medical applications, including studies of blood flow [39] and brain activity [1]. Inagaki et al. [23] successfully applied thermal image velocimetry (TIV) to measure the airflow on a wall surface and over a playground with turf to study the urban thermal and wind environment on the building and street scales. Besides using TIV technique, Particle Image Velocimetry (PIV), which uses the timesequence particle images to calculate the flow velocities, is also an efficient technique to quantitatively resolve the convective flow dynamics and turbulent eddy structures [8,15,33,40,41]. Aiming at the mentioned research gaps, this study addressed the challenging problems for the natural convection from an isolated square heated surface in the stably stratified environment, with the focuses of near-surface turbulent structures and convective flow fields. This study is expected to not only fundamentally reveal the near-surface flow/heat transfer features over a square area in the stable environment but also help for better understanding the convective flow dynamics of UHIC from an ideal 2-D square city. We novelly employed a reduced-scale water tank model with a salt-water-simulated stable stratification for modelling the stable atmospheric stratification. Similarities between the reduced-scale water tank model and the prototype of the atmospheric scale flow were guaranteed with the verified criteria used in [26,6,12]. TIV and PIV techniques are applied to study the mean flow fields, turbulent flow structures, and heat transfer characteristics due to the square heated surface in such a stable environment. Simultaneously, correlation and similarity of the TIV and PIV measurements are analyzed and quantitatively compared. Details of the methods and experimental setups are introduced in Section 2. The results are presented and discussed in Section 3. Our concluding remarks are summarized in Section 4.
2. Methods A reduced-scale idealized city model was embedded in a 0.4 0.4 0.5 m water tank (Fig. 1a). A square polyimide film heater was attached to a thin copper plate (120-mm sides; 0.2 mm deep) to simulate an ideal square urban area. As is illustrated in Fig. 1b, the heater (0.1 mm deep) and copper plate (0.2 mm deep) were thin to allow the temperature fluctuation above the copper plate to penetrate them and to be captured by the infrared camera below. The walls of the water tank were made of glass to provide good access for the laser sheet for application of PIV. The bottom was an acrylic plate for purposes of heat insulation. An infrared camera (FLIR SC660, FLIR Systems Inc., Wilsonville, OR) was used to capture the heater’s surface temperature (Fig. 1b). The resolution of the infrared camera is 640 480 pixels. Movement of the eddy structures follows the mean buoyancy-driven
flow above the copper plate in the near-surface region. Therefore, cross-correlation of two consecutive temperature fields (i.e., via TIV) can provide information on the mean velocity field by tracing the movement of the temperature fluctuations. The velocity fields on horizontal planes at various levels above the urban area were measured with PIV. The PIV system (Dantec Dynamics A/S, Denmark) consists of a high-speed camera (Speed Sense M140 with a resolution of 2560 1600 pixels) and a 10-W 532-nm continuous green laser (RayPower). The seeding particles for PIV application were 20-mm polyamide particles. Both the infrared camera and high-speed camera were operated at 30 Hz. Infrared images (i.e., TIV) and particle images (i.e., PIV) were analyzed with the same software (DynamicStudio; Dantec Dynamics A/S). The adaptive PIV algorithm was chosen with a certain range of interrogation window sizes (from 16 16 to 64 64 pixels, with an adjusting grid step size of 8 8 pixels). Based on the uncertainty estimation method described by Prasad et al. [34], the error of the PIV measurements in our application is less than 6%. The mean velocity fields obtained by TIV and PIV were quantitatively compared by calculation of the cross-correlation coefficients of speed matrices. The mean speed data obtained by TIV and PIV on horizontal planes are stored in matrices MT and MP, respectively. The cross-correlation coefficients c are obtained by Eq. (4).
cmax ¼ max½cðx0 ; y0 Þ
8 9 > > > < P MP ðx; yÞ MPx0 ;y0 M T ðx x0 ; y y0 Þ M T > = x;y ¼ max rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P 2 2> > > M P ðx; yÞ M Px0 ;y0 M T ðx x0 ; y y0 Þ M T > : ; x;y
x;y
ð4Þ where MPx0 ;y0 is the mean of MP ðx; yÞ in the region under the MT, and MT is the mean of matrix MT. The maximum correlation coefficient cmax in cðx0 ; y0 Þ represents the best fit between MT and MP. The power supplied to the heater was 64 W, so the average heat flux H0 over the urban area (a square region with 12-cm sides) was 4444 W m2. The bottom surface temperature of the heater, which was around 40.2 °C during the experiments, was measured with the infrared camera. To avoid any influence on the capture of temperature fluctuation, no thermal couple was used to measure the temperature of the upper surface (the interface of the water and the copper plate). Instead, it was represented by the bottom surface temperature (T w ¼ 40:2 °C) because the heater and copper were thin (0.3 mm in total) and the temperature difference between those two surfaces was small. The ambient water temperature was T 1 ¼ 18:2 °C, which was obtained with a microscale conductivity and temperature instrument (MSCTI; PME, Inc., Vista, CA; the temperature sensor is an FP07 thermistor). The room air and surrounding environment surface temperature Ta was the same as the water temperature. A 10 10 cm observation window, as is shown in Fig. 1b, provides access for the infrared camera to the heater. Therefore, heat leaks through this window into the air via natural convection and longwave radiation. Because the bottom of the heater faces downward, the air is hotter near the heater and thus forms a stable layer. In this situation, the heat leakage due to natural convection is estimated by conductive heat transfer in the air. In this case, the heat leakage from conductive power is Pc ¼ ka ADT=Dz, where ka ¼ 0:026 W m1 K1 is the conductivity of air, and DT can be estimated as T w T a = 22 °C. Dz can be estimated as the thickness of the bottom, where a pool of stagnant air was held in that cavity just below the heater (Fig. 1b). P c can thus be calculated and has an order of 0.3 W. The heat leakage from radiative power can be calculated by Pr ¼ AerðT w T a Þ4 based on the Stefan-Boltzmann law, where A = 0.01 m2 is the area of the
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observing window,
e = 0.8 [32] is the emissivity of the polyimide
film heater, r ¼ 5:67 108 W m2 K4 is the Stefan-Boltzmann constant, and T w and T a are the surface temperature of the polyimide film and its surroundings, respectively, in Kelvins. The radiative power Pr was around 104 W. Therefore, P c and P r were significantly smaller than the power supplied to the heater and
can thus be neglected. The supplied power can be regarded as being totally released through the upper surface of the copper plate into the salt water. Stable stratification was achieved with salt water, using the method described by Fan et al. [10], to simulate the static stable background conditions in the atmospheric boundary layer. The
Fig. 1. Experimental setup. (a) Schematic of the water tank. (b) Detailed schematic of the water tank bottom in cross section. (c) Background density profile.
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Fig. 1 (continued)
density profiles were measured with the MSCTI. The measured background density profile is shown in Fig. 1c. Based on the criteria in Lu et al. [26] and Catalano et al. [5], the similarity is achieved in our experiments between the experiment setup and the urban heat island circulation (UHIC) over an idealized square urban area. The linear fit of the density data, shown in Fig. 1c, gives a density gradient dq=dz of 46.5 kg m4, with an R-square of 0.9987. The background buoyancy frequency is thus calculated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N ¼ g q dq=dz = 0.675 s , where g is the gravity acceleration, 0
and q0 is the reference density. The convective velocity scale is given as uD ¼ ½gbDH0 =ðq0 cP Þ1=3 [26,6,5], where g is the gravity acceleration, b is the thermal expansion rate of water, H0 is the surface heat flux in the urban area, and q0 and cP are the reference density and specific heat capacity of water, respectively. uD has an order of 6:87 103 m s1. The Froude number Fr ¼ uD =ðNDÞ = 0.0847, which satisfies the similarity criteria based on studies of Lu et al. [26] and Fan et al. [10]. Therefore, the results in this study could help to better understand the fundamental large-scale urban-heat-island circulation produced by an ideal square city.
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Two types of eddy structure behaviors were identified: translation (Fig. 2a) and deformation (Fig. 2b and c). The growing, shrinking, and twisting eddy structures can be observed in the deformation process shown in Fig. 2b and c. The twisting and translation of eddy structures are caused by the mean flow, which is maintained by the difference in pressure between the heated urban area and the surrounding rural area. The same phenomenon was observed by [40] and Fan et al. [11] as ‘‘bent over plumes” in the convective boundary layer from a vertical cross-section perspective. Shrinking and growing eddy structures indicate the surface renewal process [14,24,17], which is a widely studied phenomenon of heat and mass transfer at the interfaces, such as between seawater and air or between a solid surface and the adjacent fluid layer. Both hot and cold eddy structures present growing and shrinking characteristics. The growing hot eddy structures (Fig. 2b) indicate that the fluid is heated by the surface and that the thermal boundary is becoming thicker, forming a larger thermal plume. In contrast, the shrinking hot eddy structures are caused by the rising and detachment of hot fluid parcels from the surface, which results in the invasion of cold surrounding fluid into the original position of the thermal plumes. The impact of cold fluid parcels on the surface can lead to the growth of the blue eddy structures. The shrinking blue eddy structures are caused by heat from the hot surface, which results in the gradually decreasing sizes of cold fluid parcels. These identified characteristics of eddy structures can be applied to model sensible and latent heat flux and cloud formation over urban and rural areas. These behaviors of eddy structures are also important for applications on heat flux measurement using a surface renewal model. It has great potential on heat flux measurements over the earth’s surface to predict climate change and on the heat exchangers of refrigerators or boilers to improve the efficiency of energy use and to study the mechanism of heat transfer on surfaces. The bulk heat transfer coefficient over the heated surface in the current experimental setup is also obtained quantitatively based on the parameters given in Section 2. In our study, the Nusselt number under the stable stratification Nus can be calculated to be 40 according to Eq. (3) while the Rayleigh number Ras is given as 7:7 108 based on Eq. (2). With the same conditions but in the neutral environment, the Nusselt number Nun is obtained as 1=3 ). Therefore, the relative differ140 (Nun ¼ 0:152 7:7 108 ence of Nuesselt numbers under different conditions has an order of ðNun Nus Þ=Nun 100% ¼ 71% . It is shown that the heat transfer coefficient in a stable environment is significantly lower than that in a neutral environment by 71%.
3. Results and discussion 3.1. Eddy structures and heat transfer
3.2. Analysis of velocity fields obtained by PIV
Eddy structures over the heated urban area were visualized by mapping the temperature fluctuation field. The characteristics of those eddy structures are shown in Fig. 2. The frames in Fig. 2 were taken when the flow reached a quasisteady state. Fig. 2a(i) and b(i) are the same figure, but they were drawn separately to clearly demonstrate different types of eddies. The legends are fixed between 0.5 °C and 1.2 °C. Red1 and blue represent positive and negative temperature fluctuations, respectively. In the following text, we name the eddy structures shown by positive and negative temperature fluctuation as hot eddy structures and cold eddy structures, respectively. Fig. 2a(i) and (ii) show the temperature fluctuation fields after heating for 121 s and 123 s, respectively. The filter was a 30-s mean temperature field.
As is shown in Fig. 3, the mean velocity fields on horizontal planes were measured by PIV at various heights. The contour in Fig. 3 represents a 30-s mean speed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (s ¼ u2 þ v 2 ) on horizontal planes, where u and v are the horizontal velocity components in the x and y directions, respectively. Arrows represent velocity vectors. The horizontal and vertical axes are non-dimensional coordinates in the x and y directions respectively. D is the side length of the heated urban area. The geometric center of the urban area is set as the coordinate origin (0, 0). The acquisition frequency was 30 Hz and the duration 30 s for each data set. Fig. 3 shows the 30-s average results. It can be seen that the flow above the urban area is characterized as diagonal inflow at the lower levels (2, 4, and 6 mm) and as side outflow perpendicular to the edges at the upper level (12 mm), which supports the findings of Fan et al. [10,11]. The inflow region is thicker in the vertical direction along diagonals
1 For interpretation of color in Figs. 2 and 6, the reader is referred to the web version of this article.
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Fig. 2. (a) Translation, and (b and c) deformation, including growing, shrinking, and twisting, of eddy structures. Refer to the video in the supplementary material for a clearer demonstration of eddy dynamics.
and thus creates a ‘‘ditch-like region” between two diagonals. The strong diagonal inflows produce high-pressure ridges, and relatively low flow-resistance regions then form on four side regions,
resulting in outflow jets that resemble a ‘‘four-leaf clover.” It should be noted that the convergent center does not coincide with the geometric center of the heated urban area in this study (Fig. 3a)
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Fig. 3. Mean velocity fields over heated urban area measured by PIV on horizontal planes (a) 2 mm, (b) 4 mm, (c) 6 mm, and (d) 12 mm above the level of the heated surface. White dashed squares mark the urban area’s location.
for two possible reasons. One is that the thin copper plate (0.2 mm) is deformed under the pressure of salt water and thus forms a small incline. This inclination helps the convergent center to shift uphill akin to the UHIC on a slope [30]. The other reason is the heater’s lack of uniformity due to the manufacturing process. The characteristic of diagonal inflow is an advantage for comparison of the PIV and TIV results because it is easy to recognize. The maximum speed over the urban area has an order of 4 103 m s1. The mean velocity obtained by TIV represents the eddy movement near the surface. Therefore, it will be mainly compared with the PIV results at 2 mm (Fig. 3a) in Section 3.3. The standard deviations of the u and v components are plotted in Figs. 4 and 5, respectively. The contours in Figs. 4 and 5 represent the standard deviations of u or v velocity components. Velocity vectors (arrows) are also plotted to provide information on the velocity field at corresponding heights. The maximum standard deviation of the velocity components is around 1.2 103 m s1, which is of the same order as that of the mean speed (4 103 m s1). The standard deviations of the velocity components also present larger values along the diagonals at lower levels and in the side regions at upper levels. However, there is a different phenomenon than that of the mean speed field. As is shown in Figs. 4b, c and 5b, c, the standard deviation field of the velocity components also presents a ‘‘four-leaf
clover” shape, even those actually on lower levels in the inflow region (4- and 6-mm heights). However, the ‘‘four-leaf clover” shape only presents at the upper level (Fig. 3d; 12-mm height) for contours of the mean speed. This indicates that the outflow jets can influence the flow at 4- and 6-mm heights. High standard deviation values, which indicate a highly turbulent region, appear in the regions where the mean speed is high. It is interesting to observe that the standard deviation of the u velocity component is greater in the x direction than that in the y direction (Fig. 4d). Similarly, the standard deviation of the v velocity component demonstrates the same trend (Fig. 5d), which is attributed to the main flow direction (i.e., high fluctuation in the main flow direction). 3.3. Comparison of TIV and PIV data Fig. 6 shows the mean speed calculated by the TIV method, based on sequential infrared images. To compare the TIV data with those of PIV, the 1-s average velocity field obtained by PIV at the 2mm height is plotted in Fig. 7. As shown in Fig. 6, the characteristics of the main diagonal inflows can be successfully captured by TIV calculation. However, the TIV and PIV results have two main different aspects. One is that the short-term average velocity fields obtained by TIV (1-s, 10-s,
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Fig. 4. Standard deviations of the u velocity component at heights of (a) 2 mm, (b) 4 mm, (c) 6 mm, and (d) 12 mm, respectively.
and 30-s average fields shown in Fig. 6a, b, and c respectively) are less smooth than those obtained by PIV (Fig. 7). The diagonal inflow cannot be observed at all in the 1 s-average velocity field obtained by TIV (Fig. 6a). The other aspect is the magnitudes of the measured speed. The maximum speed over the urban area measured by TIV has an order of 4 105 m s1, which is smaller than that measured by PIV by two orders of magnitude. There are several reasons for the differences in the measurement principles for PIV and TIV. First, the temperature fluctuation captured by TIV does not necessarily represent the movements of the eddy structures at the 2-mm height, where the velocity field is measured by PIV. Those captured temperature fluctuations reflect movements of the eddy structures near the surface [19,20,23], but it is difficult to identify the exact height. According to Puthenveettil and Arakeri [35], the thermal boundary layer depth dt has an order of Rac , where c (0.28–0.35 in [27] 0.309 in [7] 0.29 in [3] 0.288 in [37] is a coefficient and Ra ¼ gbDTD3 =ðmaÞ is Rayleigh number. D = 12 cm is the side length of the urban area. g is the acceleration of gravity, and b is the thermal expansion rate of water. DT ¼ T w T 1 is the temperature difference between the heated surface and the ambient fluid. a is the thermal diffusivity of water, and m is the kinematic viscosity. Therefore, the Ra in our experiments had an order of 7:6 108 . The thermal boundary layer depth dt can thus be estimated based on the coefficients mentioned above and ranges from 0.78 to 3.26 mm. Because the copper plate’s
surface is a no-slip boundary, the velocity at the surface is zero. Therefore, movements of those visualized eddy structures represent the flow between 0 and 0.78 or 3.26 mm above the copper plate. Second, the movements of the large eddy structures are slower than the instantaneous background flow. Small eddy structures can follow the main flow well, but their temperature fluctuations are not sufficiently large to penetrate the heater and copper plate and be captured by the infrared camera. This means that small (i.e., not large enough to penetrate) turbulent thermal structures are difficult to measure with the TIV method and that the average time should be large to get a more representative mean flow field. An average time of 30 s is needed, as shown in Figs. 8 and 9; this will be analyzed later in detail. Although the heater and copper plate were as thin as possible, small eddy structures are still filtered. Large eddy structures are successfully measured. The dynamic interaction between large thermal plumes and the mean background flow was studied by Hunt [50] and by Fan et al. [11], and their studies suggested that plumes can be bent over by the mean background flow. Thermal plumes can also be advected by the mean flow, but the movement speed of the plume structures along the main flow direction is lower than that of the main flow. Third, the quantity of large eddy structures is not sufficient, leading to a lower time-averaged speed. If no movement of large eddy structures is captured by the infrared camera during a certain
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Fig. 5. Standard deviations of the
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v velocity component at heights of (a) 2 mm, (b) 4 mm, (c) 6 mm, and (d) 12 mm, respectively.
time slot at a certain location, the mean flow calculated by TIV would be low or even zero at that specific time slot and location. Therefore, when the speed is averaged over time, the mean speed would be low due to those small values. This phenomenon is supported by the results that a larger average time leads to a lower mean speed, as shown in Fig. 6, in which the area with a higher speed (red regions) decreases as the average time increases (from Fig. 6a to d). Although the magnitude of the flow field obtained by TIV differs from that of PIV, the characteristics of the flow field are similar. For a quantitative comparison of the PIV and TIV mean speeds, crosscorrelation coefficients were calculated based on Eq. (4), which describes the similarity between two sets of data. The detailed parameters of PIV and TIV cases are summarized in Tables 1 and 2, respectively. The cross-correlation coefficients between different TIV datasets and different PIV datasets are shown in Fig. 8. In Fig. 8, the horizontal axis is the average time (range, 1–180 s) of the TIV datasets (TIV1-TIV12), and the vertical axis shows the cross-correlation coefficients of TIV with the corresponding PIV data sets. The data show that the cross-correlation coefficients increase quickly with the average time on the left hand of the vertical dashed line and then change slowly when the average time exceeds 30 s. This trend indicates that 30 s is a proper value to calculate the mean velocity field with the TIV method. The average
time of 30 s is also supported by scaling analysis. The time scale of UHIC in our experiments is t = D/s, where D = 120 mm is the side length of the urban area and s is the convective speed, which has an order of 4 103 m s1, shown in Fig. 3. Therefore, t = 120 103 m/4 103 m s1 = 30 s. Another phenomenon is that only the cross-correlation coefficients between the TIV data sets and the PIV2mm data sets are larger than 0.75 (above the horizontal dashed line). This result suggests that the mean velocity field calculated by TIV represents the mean flow at or below the 2-mm height. The results of PIV2mm1s and PIV2mm, shown in Fig. 8, are similar, indicating that a 1-s average in the PIV calculation can represent the mean flow well. This conclusion is also supported by the cross-correlation coefficients between different PIV data sets, as shown in Table 3. As shown in Table 3, the cross-correlation coefficient between PIV2mm1s and PIV2mm is 0.995, which is rather high and explains the similarity between PIV2mm and PIV2mm1s in Fig. 8. This further verifies that the 1-s average of the velocity field in the PIV calculation can represent the mean flow well in a water tank model of UHIC. The cross-correlation coefficient between PIV4mm and PIV2mm is also relatively high (0.957) because both are located in the inflow region of the UHIC at lower levels, and both present similar characteristics of main diagonal inflow. In contrast, the totally different flow types between PIV2mm (diagonal inflow) and PIV12mm (side outflow) lead to a low correlation coefficient
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Fig. 6. Mean speed calculated based on infrared images with averaging time of (a) 1 s, (b) 10 s, (c) 30 s, and (d) 180 s. White-filled arrows indicate the main flow directions.
Fig. 7. (a) 1-s average velocity field over heated urban area measured by PIV at the 2-mm height. (b) Enlargement of velocity field, marked by black dashed square in (a), to show the same scope of TIV view. White-dashed square indicates the location of the heated square area.
(0.67). The cross-correlation coefficients between PIV and TIV for different time slots are plotted in Fig. 9. As shown in Fig. 8, 30 s is a proper value to calculate the mean velocity in the TIV application. Therefore, consecutive data sets
with a 30-s interval were used in Fig. 9. No obvious increasing or decreasing trend for coefficients is seen in Fig. 9, which indicates that the flow reaches a quasi-steady state after 120 s of heating. The coefficients are high (above the horizontal dashed line) only
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Y. Fan et al. / International Journal of Heat and Mass Transfer 140 (2019) 660–670 Table 2 Parameters for TIV data (acquisition frequency, 30 Hz).
Fig. 8. Cross-correlation coefficients between different TIV datasets and different PIV data sets. Vertical and horizontal blue dashed lines mark the average time of 30 s (TIV7) and the cross-correlation coefficient of 0.75, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
TIV datasets
Time slots of datasets
Duration and average time (s)
TIV1 TIV2 TIV3 TIV4 TIV5 TIV6 TIV7 TIV8 TIV9 TIV10 TIV11 TIV12 TIV13 TIV14 TIV15 TIV16 TIV17
120–121 s 120–122 s 120–123 s 120–125 s 120–130 s 120–140 s 120–150 s 120–180 s 120–210 s 120–240 s 120–270 s 120–300 s 150–180 s 180–210 s 210–240 s 240–270 s 270–300 s
1 2 3 5 10 20 30 60 90 120 150 180 30 30 30 30 30
Table 3 Cross-correlation coefficients among various PIV data sets.
PIV2mm1s PIV2mm PIV4mm PIV6mm PIV12mm
PIV2mm1s
PIV2mm
PIV4mm
PIV6mm
PIV12mm
1 0.995 0.957 0.796 0.67
0.995 1 0.956 0.796 0.66
0.957 0.956 1 0.88 0.68
0.796 0.796 0.88 1 0.62
0.67 0.66 0.68 0.62 1
4. Conclusions
Fig. 9. Cross-correlation coefficients between 30-s mean speed calculated by PIV and that calculated by TIV.
Table 1 Parameters for PIV data (acquisition frequency, 30 Hz). Note that PIV2mm, PIV4mm, PIV 6 mm, and PIV 12 mm were not adopted in the same time slot because velocity fields at different levels cannot be obtained simultaneously and must be measured in sequence. Heating began at 0 s. PIV datasets
Time slots of data sets
Duration and average time (s)
Distance between measuring plane and surface (mm)
PIV2mm1s PIV2mm PIV4mm PIV6mm PIV12mm
120–121 s 120–150 s 225–255 s 332–362 s 442–472 s
1 30 30 30 30
2 2 4 6 12
for those with PIV measurements at the 2-mm height (PIV2mm and PIV2mm1s), which also suggests that the TIV measurements in our study are more representative for mean flow fields 2 mm or less above the surface.
In this study, TIV and PIV are applied in investigations of the natural convection from an isolated square heated surface in the stably stratified environment. Two different behaviors of eddy structures, including translation and deformation, were observed, along with three types of deformation processes: growing, shrinking, and twisting. These identified behaviors are important for heat and mass transfer studies in the boundary layer over surfaces and have great potential applications in the measurement of heat flux over the earth’s surface and in heat exchangers. The bulk heat transfer coefficient in a neutral environment is found to be higher by 250% than that in the studied stable environment. Besides, the mean flow field and turbulent characteristics on different horizontal planes were further measured by PIV. The results suggest that the asymmetric flow structure over a square urban area, which is characterized as the main diagonal inflow at lower levels and as the side outflow (perpendicular to edges) at upper levels. The TIV technique depends on the transient temperature variations transferred by the flow eddies. The limitation is that small turbulent eddy structures can hardly be captured by TIV because small temperature fluctuations are not significant enough to penetrate the heater and the copper plate. Therefore, a longer average time (30 s in this study) is required for TIV application to obtain a typical mean flow field. In comparison, a 1-s average is enough for PIV measurements because the PIV technique directed measure the seeding particle displacement with the flow motions. The mean speeds over the square heat surface from PIV and TIV are compared by using cross-correlation coefficients to quantify the similarity. The mean speed measured by TIV with an average time above 30 s has a high correlation (above 0.75) with that measured by PIV at 2 mm above the surface. The low-level (2 mm or less above) velocity field measured by TIV also well presents the similar characteristics of near-surface diagonal inflow observed in the PIV measurements. These similar results verify the capability of TIV for resolving the near-surface convective flow fields.
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