Introduction

The natural convection along building walls, i.e., wall flow (Fan et al. 2016; Fan et al. 2017b), is important for ventilating a city and building energy balance when there is no synoptic wind (Barlow et al. 2004; Rotach et al. 2004; Hernandez et al. 2003; Ghiaus 2013; Naveros et al. 2014). As cities get denser and higher, the wind is also blocked by the dense high-rise buildings in the urban area (Macdonald 2000; Belcher et al. 2003; Coceal and Belcher 2004; Di Sabatino et al. 2008). The energy balance in the building walls can affect outdoor/indoor thermal comfort, building energy consumption, pollutants dispersion (Yang et al. 2020; Zhang et al. 2019; Hang et al. 2018) and urban ventilation (Zhao et al. 2020; Li et al. 2022; Zhao et al. 2021). The urban radiative heat flux can also be affected by building walls and contribute to the urban heat island or urban cool island effects (Yang and Li 2015; Yang et al. 2012; Yang et al. 2017), which affects the urban heat dome flow (Fan et al. 2017a; Fan et al. 2018a; Fan et al. 2018b). Many researchers have studied the energy balance models on walls (Ghiaus 2013; Naveros et al. 2014) or the ground (Liebethal and Foken 2007). To study the energy balance in the urban canopy layer, the mechanism of the wall flows caused by natural convection must be also well understood.

Natural convection along a vertical heated plate (wall flows) has been studied for a long time such as (Eckert and Jackson 1950; Ostrach 1952; Cheesewright 1968; Tsuji and Nagano 1988; Abu-Mulaweh et al. 2000). However, the flow and heat transfer along a real high-rise building wall caused by natural convection was rarely measured due to the difficulties in finding a proper site and getting permission on taking measurements, the field measurement results are of great significance to better study the wall flow. It is also tricky to install the measurement devices adjacent to the wall without doing damage to the building wall surfaces.

Therefore, this paper aims to study the wall flow mechanism and wall energy balance caused by natural convection, by measuring the wall flow, surface temperature and radiation along a high-rise building in the field on calm clear days, allowing a meaningful discussion of the energy balance of urban canopy. Wells and Worster (2008) modeled the wall flows with a two-layer boundary layer structure when the Rayleigh number can reach as large as 1020. The experiment results of references (Fan et al. 2016; Fan et al. 2017b) support the two-layer boundary layer theory. Based on the two-layer boundary layer theory, a model was proposed by (Fan et al. 2016; Fan et al. 2017b) to calculate the convective heat transfer and airflow along the high-rise building walls. The energy balance can then be evaluated with a given natural convective heat flux on the external surface of the building wall. This study carried out experiments at very high Ra number conditions on a high-rise building, which would be useful for designing high performance low carbon buildings in the future. A method is also developed and applied to visualize airflow with infrared videos. The energy balance on a real high rise building is measured and analyzed, which is helpful for the validations of numerical models.

The paper is organized as follows. Measurement sites and methods are introduced in Section “Measurement and methods”. Results are given in Section “Results” Finally, conclusions are drawn in Section “Conclusions”.

Measurement and methods

The field measurement is carried out at a 16-story building, which is located in the campus of Sun Yat-Sen University (N 23°06′15″; E 113°17′42″), Guangzhou. The building was still under construction during the experiment in 2014-2015, so the impact of air conditioning and any other anthropogenic heat in the building was minimized. Fourteen 3D ultra-sonic anemometers [Wind Master 3D (20 Hz) and R3-50 (50 Hz), Gill Instruments, Lymington, UK] were installed on an extended horizontal rod normal to the wall on three floors. Infrared videos [Infrared camera: FLIR SC660, FLIR Systems Inc., Wilsonville, Oregon] of the tested wall were taken every hour at a frequency of 30 Hz for 5 minutes during the whole measurement period to get the wall surface temperature and temperature fluctuation. The surface temperature of east facing wall, south facing wall, west facing wall, north facing wall and the building roof were recorded by thermal couples continuously every 1 minute. Solar radiation intensity was recorded by weather stations [PortLog-Weather station, RainWise Inc., Trenton, Maine]. The smoke was set off in the previous test (30th, July 2014) to visualize the wall flows, but no smoke was released in this in-situ measurement due to the permission issue and difficulty in finding a time period when there is no synoptic wind (we cannot predict when there is no wind. There are many eddies in the atmosphere that can destroy the smoke flow along the wall).

The energy balance on the south facing wall surface was calculated based on the energy conservation, which is shown in the Fig. 1.

Fig. 1
figure 1

Heat flux through the wall surface. Heat conduction = Solar radiation - Natural convection - Long wave radiation

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The surface temperature was measured by both infrared camera and thermal couples. The uniformity of the wall surface temperature can be shown in Fig. 2.

Fig. 2
figure 2

Infrared picture of the south facing wall. It was taken at 12:00, on 15th October. The position of thermal couples is marked on the picture

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The wall surface temperature is uniform, so we can use the average temperature measured by the thermal couples to represent the wall temperature. All the thermal couples were calibrated by constant temperature water tank, so the data measured by thermal couples would be more accurate. Only two levels can be observed due to the limitation of the IR camera scope. The two levels shown in the figure are from 10th to 14th floor. The calculation of solar radiation heat flux, natural convection heat flux and long wave radiation heat flux are shown in the following subsections.

Solar radiation

The position of the sun relative to the horizontal surface can be described by the solar zenith angle θz, solar elevation angle α and solar azimuth angle ϕ, which are illustrated in Fig. 3.

Fig. 3
figure 3

Illustration of solar zenith angle θz, solar elevation angle α and solar azimuth angle ϕ

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The method of obtaining the solar radiation on the south facing vertical wall is shown in Fig. 4.

Fig. 4
figure 4

The method of calculating solar radiation on the vertical surface

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Transmittance is the atmosphere transmittance of solar radiation. The first step is to calculate transmittance by using two known parameters: solar radiation given off by the sun in the horizontal direction and the solar radiation received by the weather station on the horizontal surface. The second step is to calculate solar radiation received by the vertical surface by using two known parameters: solar radiation given off by the sun in the vertical direction and transmittance. See text for the detailed calculation.

Solar radiation is the sum of direct solar radiation Idir and diffuse irradiance Idif. The calculation of Idir and Idif are shown in Eqs. (1)-(2) given by Bouguer’s law.

$${I}_{dir}={I}_0{E}_0{P}^{1/\sin \alpha}\cos \theta$$
(1)
$${I}_{dif}=0.5{F}_{bs}{I}_0\sin \alpha \frac{1-{P}^{1/\sin \alpha }}{1-1.4\ln P}$$
(2)

where I0 is solar constant with a value of 1367 W m− 2, which is recommended by Fröhlich and Brusa (1981). E0 = 1 + 0.033 cos(2πdn/365) is defined by Duffie and Beckman (1980) as the eccentricity correction factor of the earth’s orbit and dn is the day number of the year from 1st January. P is the atmosphere transmittance of solar radiation; α is solar elevation angle (also called solar altitude). θ is solar incidence angle. Fbs is shape factor between the calculated surface and sky.

The definition of solar incidence angle is given in Eq. (3).

$$\cos \theta =\cos \zeta \cos {\theta}_z+\sin \zeta \sin {\theta}_z\cos \left(\phi -\gamma \right)$$
(3)

where ζ is the slope of the calculated surface (ζ = 0 when the surface is horizontal and ζ = 90° when the surface is vertical); θz is solar zenith angle; ϕ is solar azimuth angle; γ is the azimuth angle of the calculated surface (γ = 0 when the surface is facing south, γ = 90° when the surface is facing east and γ = − 90° when the surface is facing west).

The solar zenith angle θz, solar altitude α and solar azimuth ϕ (ϕ = 0 when the sun is in the south, ϕ = 90° when the sun is in the east and ϕ = − 90° when the surface is in the west) are obtained in Eqs. (4)-(6).

$$\cos {\theta}_z=\cos \delta \cos \varphi \cos h+\sin \delta \sin \varphi$$
(4)
$$\alpha =90-{\theta}_z$$
(5)
$$\cos \phi =\frac{\sin \alpha \sin \varphi -\sin \delta }{\cos \alpha \cos \varphi }$$
(6)

where δ = 23.45 sin[360 × (dn + 284)/365] is solar declination; φ is the geographic latitude in degrees (the North Hemisphere is positive and the South Hemisphere is negative); h = 15 × (12 − hr) is hour angle (hr is the local time value, e.g. hr =12 at noon and hr =18 at 6 o’clock afternoon).

All the variables to calculate the solar radiation are known except for P, the atmosphere transmittance of solar radiation. Fortunately, the solar radiation on the horizontal surface was measured by weather stations, so the transmittance can be calculated by the following Eq. (7).

$${I}_{measure}={I}_{dir}+{I}_{dif}={I}_0{E}_0{P}^{1/\sin \alpha}\cos \theta +0.5{F}_{bs}{I}_0\sin \alpha \frac{1-{P}^{1/\sin \alpha }}{1-1.4\ln P}$$
(7)

The latitude of the measurement point is N 23°06′15″. The experiment was carried out from 14th October to 16th October, 2014. The weather station was set on the roof of the 16−story building and the surrounding area is relatively open, so the shape factor Fbs can be assumed to be 1 for the weather station location.

After the transmittance is solved, the short wave radiation on the south facing vertical wall Isv can be calculated by Eq. (8).

$${I}_{sv}=\left(1-a\right)\left({I}_{dirv}+{I}_{difv}\right)$$
(8)

where a is albedo. Idirv and Idifv are direct radiation and diffuse radiation on the south facing vertical wall respectively.

Long wave radiation

The long wave radiation can be divided into two parts. First, the longwave radiation between the vertical wall and sky and Second, the longwave radiation between the vertical wall and other ground surfaces. To calculate the long wave radiation between the vertical wall and sky, the sky temperature needs to be modeled. Berdahl and Martin (1984) sky temperature model has been widely used, which is shown in Eq. (9).

$${T}_{sky}={T}_a{\left[0.711+0.0056{T}_{dp}+0.000073{T}_{dp}^2+0.013\cos (15t)\right]}^{1/4}$$
(9)

where Tsky (K) is sky temperature; Ta (K) is ambient dry bulb temperature; Tdp (°C) is dew point temperature; t is the hour from midnight.

The long wave radiation Ilw of the vertical wall can be calculated based on the Stefan-Boltzmann law by the Eqs. (10)-(12).

$${I}_{lw}={I}_{lbs}+{I}_{lbg}$$
(10)
$${I}_{lbs}={\sigma \varepsilon}_b{F}_{bs}\left({T}_{sky}^4-{T}_b^4\right)$$
(11)
$${I}_{lbg}={\sigma \varepsilon}_b{\varepsilon}_g{F}_{bg}\left({T}_g^4-{T}_b^4\right)$$
(12)

where σ = 5.67 × 10−8 W m− 2 K− 4 is the Stefan-Boltzmann constant. εb and εg are emissivity for the test building wall surface and surrounding ground respectively. The emissivity of the surrounding surfaces is assumed to be uniform. Ilbs is long wave radiation heat flux between the vertical wall and sky. Ilbg is long wave radiation between the vertical wall and other surfaces, which is calculated based on simplified equation for the non-enclosed envelope. The tested building is much higher than its surroundings and located in an open area, so the shape factors Fbs (shape factor between the building vertical wall and sky) and Fbg (shape factor between the building vertical wall and the ground) are assumed to be 0.5. Tb and Tg are the surface temperature of the vertical wall and the surface temperature of surrounding surfaces respectively.

Natural convective heat transfer

The local convective heat transfer rate can be represented by the non-dimensional Nusselt number Nux, which is defined in Eq. (13).

$${Nu}_x=\frac{h_xx}{\lambda }$$
(13)

where hx is local convective heat transfer coefficient; x is characteristic length scale, which is vertical coordinate along the wall in this experiment; λ is thermal conductivity.

The local Rayleigh number Rax is defined as Eq. (14).

$${\textrm{Ra}}_x=\frac{g\beta \Delta {Tx}^3}{\nu \kappa}$$
(14)

where x is length scale. g is acceleration of gravity. β is thermal expansion coefficient. ΔT is difference between the fluid temperature and ambient temperature. ν is kinematic viscosity. κ is thermal diffusivity.

The correlation between Nux and Rax for convection is shown in Eq. (15).

$${\textrm{Nu}}_x=c{\textrm{Ra}}_x^b$$
(15)

where c and b are constants.

The background wind speed was small when the experiment was carried out. The weather was clear and there were no clouds. The convective heat transfer was dominated by natural convection, which is also confirmed in Fan et al. (2016) and Fan et al. (2017b). Therefore, the natural convection correlation between Nux and Rax can be adopted. Fan et al. (2016) and Wells and Worster (2008) suggested that the flow was in region II, where the buoyancy instability dominates and the heat flux is constant, under this experiment condition. Tsuji and Nagano (1988) and Wells and Worster (2008) supported the correlation of natural convection along a heated vertical wall in region II, which is shown as Eq. (16).

$${\textrm{Nu}}_x=0.12{\textrm{Ra}}_x^{1/3}$$
(16)

Based on Eqs. (13)-(16), the heat flux qx caused by natural convective heat transfer can be obtained as Eq. (17).

$${q}_x=-{h}_x\Delta T=-0.12\lambda {\left(\frac{g\beta}{\nu \kappa}\right)}^{1/3}\Delta {T}^{4/3}$$
(17)

As we can see in Eq. (17), the local heat flux is independent of vertical coordinate x, which means that the heat flux is constant over the wall. Negative values of qx means the heat flux is from the wall to the ambient air.

Turbulent flow visualization method by infrared videos

Fluctuations in a turbulent flow are usually classified into two broad classes. There are those fluctuations which are disorganized enough so that we have to un-specifically describe them as “turbulent.” But one sometimes finds recognizable patterns, such as groups of vortices, or vortices of special shape, which appear repeatedly. As pointed out by Haramina and Tilgner (2004), these organized recurrent patterns are “coherent structures” or “eddies” which are distinguished from the turbulent background.

Turbulent flow boundary layer has a big influence on the local heat transfer rate, which will change the local surface temperature. The turbulent velocity is time-dependent, so the surface temperature will also fluctuate with time. If the fluctuation and “coherent structures” in the boundary layer are large enough, they could be captured by the high resolution infrared camera.

The thermal image method has been adopted by many researchers, such as Hetsroni et al. (1996), Hetsroni and Rozenblit (2000), Kowalewski et al. (2000) and Hetsroni et al. (2001). Inagaki et al. (2013) gave a detailed description on how to get the velocity of the turbulent flow from thermal videos and called the method as thermal image velocimetry (TIV). The general procedure for TIV is shown in Fig. 5.

Fig. 5
figure 5

General procedure of Thermal Image Velocimetry (TIV)

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There are several methods to calculate the average value of the image sequence in step two, which are introduced in detail by Inagaki et al. (2013). The time average method is widely used in TIV. The time average method is given in Eq. (18).

$$T\left(x,t\right)=\left\langle T(x)\right\rangle +{T}^{\prime}\left(x,t\right)$$
(18)

where T(x, t) is temperature changing with position and time; 〈T(x)〉 is time average temperature changing with position; T′(x, t) is temperature fluctuation changing with position and time. The application of TIV method can be affected by the following factors: (1) the characteristics of tested surfaces (surfaces that have small thermal storage and thermal conductivity are preferred); (2) the temperature difference between the surface and ambient fluid (the temperature difference should be large enough); (3) the resolution of infrared camera; (4) the distance between the camera and the surface (the small distance is preferred. If the distance is large, the infrared ray emitted by the surface can be partially absorbed by the atmosphere between the surface and thermal camera, which will cover the fluctuation of the surface temperature).

Two kinds of set-ups are usually used in the TIV method, which is shown in Fig. 6.

Fig. 6
figure 6

Two general set-ups. a Visualization of airflow on the same side with the camera. b Visualization of water flow on the different sides against the camera

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The set-up in Fig. 6 (a) is used to visualize the airflow. The influence on hot plate surface temperature caused by the airflow is difficult to penetrate to another side of the solid plate and the infrared radiation can transmit through air, so the fluid should be between the camera and the observed surface. The set-up in Fig. 6 (b) is used to visualize the water flow because of the following reasons: (1) the infrared radiation cannot penetrate the water, so only the water surface temperature could be observed if the camera and water flow are on the same side of the plate; (2) the heat transfer rate between the water and the plate is much larger than that between the air and the plate, so the temperature fluctuation can penetrate into the plate and be observed on the other side of the plate; (3) the temperature fluctuation caused by the air is so small compared with the fluctuation created by the water that can be neglected.

The reason that calibration needs to be done in the last step is as follows. The velocity at the wall surface should be zero according to the no-slip boundary theory, so the movement shown on the surface is actually the movement of coherent structures in the boundary layer. Although the velocity of coherent structures can represent the mean velocity to some degree, they are not exactly the same. Therefore, the calibration needs to be done.

The set-up used in this field measurement is the same with the set-up in Fig. 6 (a). Due to the limitations of experiment conditions, such as large thermal storage of the wall, the long distance between the wall and the camera, only qualitative visualization was done.

Results

The surface temperature of south facing wall and ambient conditions are shown in Fig. 7.

Fig. 7
figure 7

Ambient condition of temperature and relative humidity. Wall surface temperature represents the surface temperature of south facing wall. The sky temperature is calculated by Eq. (9)

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As shown in Fig. 7, the surface temperature and air temperature have regular diurnal changes, while the sky temperature is influenced by the relative humidity, which will result in changing the long wave radiation heat flux. Therefore, the large humidity can restrain the heat transfer by long wave radiation and confine the heat in the urban canopy layer.

The solar radiation on the building roof is measured by the weather station. The measured solar radiation, calculated diffuse radiation, direct radiation and transmittance are shown in Fig. 8. The ratio of diffuse radiation to direct radiation is shown in Fig. 9.

Fig. 8
figure 8

Solar radiation on the horizontal surface. The transmittance in the figure is atmospheric transmittance of solar radiation

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Fig. 9
figure 9

Ratio of the diffuse radiation to direct radiation

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According to Idso (1969), the atmospheric transmittance is mainly influenced by three factors: the absorption and scattering by water vapor; the absorption and scattering by dust in the air; scattering by dry dust-free air. The transmittance calculated based on the measured solar radiation keeps decreasing from sunrise to sunset.

The diffuse solar radiative heat flux is larger than that of the direct radiation after 16:00 on each day, when the solar elevation angle and transmittance are very small. The direct radiation, diffuse radiation and transmittance have a big fluctuation around 12:00, 16th October (circled by dashed elliptical in Fig. 8). The fluctuation is caused by the clouds cover over the sky at that time. The cover of the clouds also shows the influence on the ratio of the diffuse radiation to the direct radiation during that period. According to Fig. 9, the ratio keeps decreasing in the morning and increasing in the afternoon. The trend agrees well with the results in Idso (1971). The reason is obvious. The travel distance for the light is the shortest at mid-noon, so the direct solar radiation reaches its maximum value.

The flow along the wall caused by natural convection is shown in Fig. 10.

Fig. 10
figure 10

Visualization of the wall flow caused by natural convection. The average data is obtained from 5 minute image sequence. The temperature shown in the figure is fluctuation after the average is removed. The frequency of videos is 30 Hz. The time interval between (a) (b) and (c) is 1.33 seconds

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The coherent structure circulated by the yellow dashed elliptical in Fig. 10 can be observed obviously moving upwardly.

The short wave radiation received by the south facing vertical wall, the heat flux given off by the wall and the conductive heat flux are summarized in Fig. 11.

Fig. 11
figure 11

Energy balance on the wall surface during clear days. The short wave radiation is the received solar radiative heat flux (including direct radiation and diffuse radiation) of the south facing wall. The long wave radiation is the long wave radiative heat flux (including the radiation between the wall and the ground and the radiation between the wall and the sky). The conductive heat flux is the heat conducted into the wall (positive heat flux) and the heat conducted out of the wall (negative heat flux)

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As observed in Fig. 11, the phases for different kinds of heat fluxes are different under clear day conditions. The conductive heat flux into the wall first reach its maximum at around 11:00, and then the short wave radiation appears the peak at around 12:00. The following is the convective heat flux caused by natural convection when the surface temperature is largest at around 14:00. The long wave radiation heat flux appears its peak in the last at around 15:00. During night time, the heat in the wall is mainly transferred out by long wave radiation. The conductive heat flux out of the wall is equal to the long wave radiation heat flux at night. Therefore, if the building density in urban areas is too large, it could reduce the long wave radiation and trap the heat in the urban canopy layer. The heat flux caused by natural convection and long wave radiation are in the same order during daytime. The slope of conductive heat flux is large in the morning due to the small slope of convective heat flux and long wave radiation heat flux. The large thermal inertia prevents the wall surface temperature increasing rapidly, leading to the small slope of convective heat flux and long wave radiative heat flux in the morning.

Conclusions

The wall flows, surface temperature and radiation along a high-rise building are measured in the field and calculated. The flow along the wall caused by natural convection is visualized by the fluctuation of the surface temperature obtained from the infrared image. The flows along the wall are dominated by natural convection during the calm clear days, so the natural convective heat transfer is used in the energy balance analysis. Solar radiative heat flux and long wave radiative heat flux are also calculated to complete energy balance model on the wall surface. The atmosphere transmittance is not a constant and keeps decreasing from sunrise to sunset during clear days. The different kinds of heat flux reach their maximum value in a day cycle at different time. The conductive heat flux first reaches the peak. Afterwards, the solar radiative heat flux gets the maximum point and then followed by the natural convective heat flux and long wave radiative heat flux. Natural convective heat flux has the same order with long wave radiative heat flux during day time. The large humidity can restrain the heat transfer by long wave radiation and confine the heat in the urban canopy layer. Since the heat conduction outside of the wall is dominated by the long wave radiation, the dense high rise buildings will trap the heat in the urban canopy layer.