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  • View in gallery

    (a) Water-tank experiments. The urban dome that is formed over an urban area (simulated by many hot copper plates and cylinders) is visualized with thermochromic liquid crystal. Stable stratification in the atmosphere is simulated with saltwater. (b) Schematic of urban-dome flow. Three possible scenarios of pollutant dispersion outside an urban area are shown when urban-dome flow is formed.

  • View in gallery

    Nondimensional horizontal velocity profiles at the urban edge. Open symbols represent water-tank-experiment models. Filled symbols are atmospheric-scale models. Wang and Li simulation 1 and Kristóf et al. (2009) are numerical simulations of water-tank experiments. The data from Lu et al. (1997b) are not from the exact urban edge: the position is r/D = 0.53.

  • View in gallery

    An illustration of the bulk model of urban-dome flow in a water-tank model (Pr > 1): Pr is Prandtl number; zi and zr are the mixed height and radial velocity reverse height, respectively; is the average velocity of inflow, also regarded as the velocity scale in the urban-dome study; is the average vertical velocity relating to the thermal boundary layer; is the average vertical velocity relating to the plume’s hydrodynamic diameter; δυ and δt are the plume hydrodynamic diameter and plume thermal diameter, respectively.

  • View in gallery

    The relationship between lυ/lt and Pr: lυ and lt are the hydrodynamic boundary layer thickness and thermal boundary layer thickness, respectively. The curve is fitted with the power function lυ/lt = CPrm, where C (=1.43) and m (=0.38) are positive constants (correlation coefficient squared R2 = 0.9981; the residual sum of squares is 0.003 04). The results are taken from Worster (1986) to demonstrate that the ratio of the hydrodynamic boundary layer thickness to the thermal boundary layer thickness changes with Prandtl number according to a power law.

  • View in gallery

    A comparison of different nondimensional radial velocity profiles considering the influence of Pr. Filled circles represent the mean nondimensional radial velocity of real-scale models of the atmosphere in Fig. 2. Open squares represent the mean nondimensional radial velocity of water-tank experiments in Fig. 2. Filled blue triangles represent the nondimensional radial velocity in water-tank experiments corrected using Eq. (21) considering the influence of the Prandtl number (i.e., the new convection velocity scale); uD is the traditional velocity scale without consideration of the Prandtl number. Water corrected by UE, UG, and UL represents the corrected nondimensional radial velocity of water-tank experiments as based on the velocity scales of Eckert and Jackson (1950), Gebhart et al. (1988), and Ostrach (1982), respectively.

  • View in gallery

    Nondimensional vertical velocity at the center of the heated area changing with a nondimensional height: (a) original data from water-tank experiments and air models, and (b) selected data for comparison with data from air models and water-tank experiments with Pr correction. See the text for a detailed explanation.

  • View in gallery

    As in Fig. 6, but for nondimensional standard deviations of the horizontal velocity components.

  • View in gallery

    As in Fig. 6, but for nondimensional standard deviations of vertical velocity components.

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A New Convective Velocity Scale for Studying Diurnal Urban Heat Island Circulation

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  • 1 Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China
  • | 2 Climate Modeling and Impacts Laboratory, Department of Sustainability, Casaccia Research Centre, ENEA, Rome, Italy
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Abstract

Urban heat island circulation establishes an urban dome under stable stratification and no background wind conditions. Small-scale water models have been a very useful tool in the exploration of the mechanisms by which urban domes and their associated wind flows are formed. Data are available from a number of water-tank heat island models. Data from field measurements, computational fluid dynamics, and small-scale water-tank experiments are compared in this paper. The small-scale water-tank experiments were found to produce relatively low radial velocities, such as the radial horizontal velocity. Different relevant velocity scales developed in the literature were reviewed. The influence of the Prandtl number on convective flows was analyzed. The analysis resulted in a new convective velocity scale that is a function of the Prandtl number, and the new scale was found to work well. This new development is expected to render small-scale models more useful in urban wind studies. The new convective velocity scale may be extended to water-modeling studies of other buoyancy-driven airflows.

Corresponding author address: Yifan Fan, The University of Hong Kong, Dept. of Mechanical Engineering, Rm. G05, Yam Pak Bldg., Pok Fu Lam Rd. 79C, Hong Kong 999077, Hong Kong. E-mail: u3002019@connect.hku.hk

Abstract

Urban heat island circulation establishes an urban dome under stable stratification and no background wind conditions. Small-scale water models have been a very useful tool in the exploration of the mechanisms by which urban domes and their associated wind flows are formed. Data are available from a number of water-tank heat island models. Data from field measurements, computational fluid dynamics, and small-scale water-tank experiments are compared in this paper. The small-scale water-tank experiments were found to produce relatively low radial velocities, such as the radial horizontal velocity. Different relevant velocity scales developed in the literature were reviewed. The influence of the Prandtl number on convective flows was analyzed. The analysis resulted in a new convective velocity scale that is a function of the Prandtl number, and the new scale was found to work well. This new development is expected to render small-scale models more useful in urban wind studies. The new convective velocity scale may be extended to water-modeling studies of other buoyancy-driven airflows.

Corresponding author address: Yifan Fan, The University of Hong Kong, Dept. of Mechanical Engineering, Rm. G05, Yam Pak Bldg., Pok Fu Lam Rd. 79C, Hong Kong 999077, Hong Kong. E-mail: u3002019@connect.hku.hk

1. Introduction

The urban heat island (UHI) effect refers to the phenomenon in which an urban area (both the air and the surfaces) is generally warmer than its rural surroundings. UHI is not only a thermal phenomenon but also a driving force for a buoyancy-induced urban breeze or circulation, especially under weak synoptic systems. The basic airflow circulation is shown in Fig. 1 using flow visualization from a water-tank model combined with a sketch. The dome circulation is characterized by rising flow in the city center with radial spread at the top and radial inflow at the bottom. The dome size appears to be larger than the city’s dimensions. The air exchange across the dome boundary is limited. The existence of radial inflow below the radial velocity reverse height can bring air pollutants beyond the city’s edge into the city. Therefore, the urban dome may have a significant effect on air quality. Field measurements have shown that the lateral dimensions achieved by the urban-emission cloud (i.e., the urban dome) are an important control mechanism for peak ozone concentrations and can be even more important than the mixing height under calm weather conditions (Banta et al. 1998). Other field studies such as those by Miao et al. (2009) and Liang and Keener (2015) revealed the existence of the urban-dome flow and its importance to urban ventilation and the dispersion of pollutants.

Fig. 1.
Fig. 1.

(a) Water-tank experiments. The urban dome that is formed over an urban area (simulated by many hot copper plates and cylinders) is visualized with thermochromic liquid crystal. Stable stratification in the atmosphere is simulated with saltwater. (b) Schematic of urban-dome flow. Three possible scenarios of pollutant dispersion outside an urban area are shown when urban-dome flow is formed.

Citation: Journal of Applied Meteorology and Climatology 55, 10; 10.1175/JAMC-D-16-0099.1

A number of methods have been used to study urban-dome circulation, including analytical (linearized models), numerical (direct numerical simulation, large-eddy simulation, and Reynolds-averaged Navier–Stokes), experimental (water tanks), and observational (field measurements) methods. The primitive governing equations can be reformulated and solved with two types of methods to obtain the flow and thermal fields in the entire urban dome. The first is the perturbation method used by Starr Malkus and Stern (1953), Vukovich (1971), Smith and Lin (1982), Lin and Chun (1991), and Baik (1992). The governing equations are linearized and simplified and then solved with proper boundary conditions. It is difficult to grasp the physics of the urban dome directly from these solutions. The second consists of numerical methods, such as those used by Yoshikado (1992), Saitoh and Yamada (2004), Catalano et al. (2012a), Falasca et al. (2016), Ryu et al. (2013), and Wang and Li (2016), in which the convective boundary layer (CBL) and urban-dome flow are simulated numerically. For numerical methods, it is difficult to consider both large city-scale flow (40–200 km) and the flow details (flow around a building of 10–100 m in width) at the same time. Wang and Li (2016) tackled this challenge by using a porous model coupled with resolution of individual targeted buildings and by integrating stable stratification of the atmosphere into the model using the coordinate transformation developed by Kristóf et al. (2009).

In addition to the field studies mentioned earlier, water-tank experiments (Poreh 1996) have also been widely used to study atmospheric flow because of the advantages of easy control of the boundary conditions and convenient data acquisition (Deardorff 1970; Lu et al. 1997b; Colomer et al. 1999; Cenedese and Monti 2003; Yuan et al. 2011; Falasca et al. 2013). Similarity criteria in both geometry and flow can be satisfied (Poreh 1996; Catalano et al. 2012a). The stable stratification condition in the atmosphere may be created in a water tank by using either of two methods: 1) a nearly linear temperature gradient created by heating at the top, maintaining good thermal insulation around the water-tank walls, and controlling the temperature at the bottom or 2) a linear density gradient created by gradually filling the tank with saltwater at a specific flow rate and concentration. The second method is probably more controllable. The flow characteristics in the laboratory-scale models can represent the large-scale flows in the prototype if the specific nondimensional parameters of the scaling models and prototype are similar.

Each of these methods has pros and cons. Field measurements can provide data from a real situation, but it is difficult to separate the effects of the individual factors that affect the flow. Water-tank modeling, including modeling of a water-tank setup by using methods of computational fluid dynamics, can be adopted to obtain the spatial distribution of the temperature and flow speed at each point in the urban dome, but results must be properly scaled. Mathematical analyses of the governing equations would simplify the problem and help to capture the main characteristics of the urban-dome flow (Starr Malkus and Stern 1953; Vukovich 1971).

Convective velocity and temperature scales have been suggested by Deardorff (1970) and by Lu et al. (1997b). These scales enable the normalization and comparison of data obtained with different methods. Catalano et al. (2012a) compared the existing data from field measurements, computational fluid dynamics, and small-scale water-tank experiments and obtained good agreement in the temperature fields, but the small-scale water-tank experiments produced much lower maximum velocities, particularly for the radial (horizontal) velocities. They suspected that the velocity scales used thus far may not be good scaling parameters and that the existing theory requires further development.

Urban morphology, building and city size, heat flux, buoyancy frequency, and fluid properties are factors that can affect the urban-dome flow. The existing velocity scales are reviewed here, followed by the development of a new velocity scale that considers the influence of the Prandtl number.

2. Convective velocity scales and their comparison

Buoyancy-driven convective flows can be divided into different categories on the basis of the aspect ratio.1 Different flow aspect ratios can lead to different scaling characteristics. The UHI-induced circulation (UHIC) has neither the very large aspect ratio (approximately 1000) of the CBL (also referred to as the planetary boundary layer) nor the very low aspect ratio (0.01–0.1) of buoyancy plumes. The radial inflow boundary layer is dominated by convective turbulence; that is, the inflows generated by the UHIC are similar to the atmospheric daytime CBL (Poreh 1996). As the urban area increases or different urban areas are merged together, the scaling characteristics of the UHIC might resemble those of the CBL, and thus a quick review of the velocity scales in both the CBL and plumes is useful.

The CBL is formed during the daytime as Earth’s surface is heated by solar radiation. The UHIC can occur during both the day and night, whereas the CBL in rural areas exists only during the day. Therefore, the UHIC and CBL interact only during the day. The CBL aspect ratio (i.e., the ratio of the horizontal length scale to the vertical length scale) may be assumed to be infinity. The horizontal length scale of the CBL usually is on the order of hundreds of thousands of kilometers. The vertical length scale is the distance between the ground and the top of the mixed layer, which is also the mixed height zi, and is on the order of 1–2 km. When the background wind and surface roughness are low, convective cells can be hexagonal and can be either open or closed. Researchers have noticed that cells usually occur over the ocean instead of over land. There are still disagreements regarding the reasons that open cells are formed at some times and closed cells are formed at others. Graham (1934) and Helfand and Kalnay (1983) proposed that open cells form when convection is driven by heating at the bottom of the clouds and that closed cells form when convection is driven by the radiational-cooling effect at the top of the clouds. Baker and Charlson (1990), Stevens et al. (2005), Rosenfeld et al. (2006), Wang and Feingold (2009a,b), and Feingold et al. (2010) suggested that aerosols and precipitation play important roles in the formation of cells.

There are some common features between convective coherent structures in the laboratory and in the atmosphere: both are caused by the instability of gravity and have similar hexagonal shapes. The aspect ratios of cells in the laboratory and in the atmosphere are very different, however. The observed aspect ratio of Rayleigh–Bénard cells ranges from 2:1 to 3:1, whereas the aspect ratio of the convective cells in the atmosphere is on the order of 20:1–30:1. Helfand and Kalnay (1983) suggested that the asymmetry of heating or cooling profiles might be the cause of the atmosphere’s large aspect ratio. Agee (1984) and Krishnamurti (1975) considered that the latent heat carried by water vapor and the large-scale motion in the atmosphere could also lead to a large aspect ratio. Feingold et al. (2010) and Krishnamurti (1975) held the opinion that the turbulent Rayleigh number, in which the kinematic viscosity and thermal diffusivity are replaced by the eddy viscosity and eddy thermal diffusivity, respectively, should be used to characterize convection in the atmosphere.

The boundary conditions of convection in the atmosphere differ from those in small-scale containers in the laboratory. For example, the top of the atmosphere is not rigid. The depth of the convection layer changes over time as a result of the entrainment effect, and the heat flux is not uniform for convection in the atmosphere, as noted by Hibberd and Sawford (1994). These boundary conditions may also result in large aspect ratios in the atmosphere. The scale is so large in the atmosphere that convection is influenced by the Coriolis force, centrifugal force, and large-scale circulation. In addition, clouds induced by convection can also affect the convection in return by changing the solar radiation that reaches the ground under the clouds.

Deardorff (1970) developed the velocity scale of the CBL as
e1
where g is gravitational acceleration, β is thermal expansion rate, zi is mixed height, and is time-averaged kinematic turbulent heat flux at the surface. The large aspect ratio of the CBL is the main reason behind of the choice of the mixed height zi for use as the length scale. This velocity scale has been used in the study of CBLs by many researchers, such as Willis and Deardorff (1974), Kaimal et al. (1976), and Zilitinkevich et al. (2006).
Plumes that have a very small aspect ratio can be regarded as point-source flows. The influence of the source diameter disappears when the flow is distant from the source. In this case, the flow can be described by the plume function. If the source is an area source and has a diameter of D, the flow will be influenced by both the source geometry and the height. List (1982) adopted (B0D2/z)1/3 as the velocity scale, where
eq1
is the buoyancy flux, D is the source diameter, z is the vertical distance between the location of interest and the source, ρ0 is the reference density, and and are the time-averaged kinematic turbulent heat flux and density flux, respectively, at the surface. Colomer et al. (1999) obtained (B0D)1/3 as the horizontal velocity scale and (B0/D2)1/3z as the vertical velocity scale, where D again is the source diameter and z is the distance away from the source.

To repeat, the UHIC has neither the very large aspect ratio (approximately 1000) of the CBL nor the very small aspect ratio (0.01–0.1) of plumes. Urban domes originate from a very large area source with a horizontal extent that is much greater than the height. Lu et al. (1997b) suggested that the self-similarity region does not exist in the large-aspect-ratio “plumes” (originally expressed as “low-aspect-ratio plumes” because of different definitions of aspect ratio) such as urban domes and that traditional plume scales cannot be used for urban-dome flow. New scaling parameters should be proposed.

The UHI phenomenon is most intense at night. The rural area begins to cool after sunset as a result of radiative cooling. Stable stratification forms to replace the mixed layer during the day in the rural area. The urban area is still warmer than the rural area because of the UHI effect, and therefore convection develops between the two areas.

According to Lu et al. (1997b), the parameters that affect urban-dome flow are the atmospheric stability, which is quantitatively represented by the background buoyancy frequency N at the rural area, the city diameter D, and the heat flux H0 of the urban area. Lu et al. (1997b) proposed the following velocity scale for use in urban-dome flow:
e2
They also suggested that the Froude number, which for this purpose is defined as Fr = uD/(ND), should be the nondimensional parameter used to scale the prototypes for urban-dome flow to water-tank experiments.
The UHIC during the day differs from that at night because the daytime flow is influenced by the mixed layer in the rural area. Cenedese and Monti (2003) showed that the temporal evolution of the temperature profiles over urban areas during the day and at night are similar, which suggests that the nighttime scaling method can be used during the day. Falasca et al. (2013) carried out an experiment in a water tank to simulate the daytime UHI-induced breeze. They measured the temperature profiles in the mixed layer and obtained a relationship between the mixed height and the Froude number. They also confirmed the similarity between the urban-dome flow at night and the UHI-induced breeze during the day. Hidalgo et al. (2010) suggested that the mixed-layer height should be used in the velocity scale instead of the urban diameter because of the domination of solar radiation during the day, although the length scale should be the urban diameter. The heat flux difference between the urban and rural areas should be used. The velocity scale during the day that was used by Hidalgo et al. (2010) is
e3
where zi is the mixed-layer height and Hu and Hr are the heat flux in the urban area and the heat flux in the rural area, respectively.

Data for nondimensional horizontal velocity profiles at the urban edge are available from a number of studies (Wang and Li 2016; Lu et al. 1997b; Catalano et al. 2012a; Cenedese and Monti 2003; Yoshikado 1992; Kristóf et al. 2009). The data are summarized in Fig. 2, and the parameters and boundary conditions are summarized in Table 1.

Fig. 2.
Fig. 2.

Nondimensional horizontal velocity profiles at the urban edge. Open symbols represent water-tank-experiment models. Filled symbols are atmospheric-scale models. Wang and Li simulation 1 and Kristóf et al. (2009) are numerical simulations of water-tank experiments. The data from Lu et al. (1997b) are not from the exact urban edge: the position is r/D = 0.53.

Citation: Journal of Applied Meteorology and Climatology 55, 10; 10.1175/JAMC-D-16-0099.1

Table 1.

Summary of the parameters used in numerical simulations and water-tank experiments. Wang and Li simulation 1 and Wang and Li simulation 2 were obtained through personal communications. The method for them is based on that of Wang and Li (2016).

Table 1.

An obvious difference can be seen between the flow simulated in a small-scale water-tank model and the flow at the atmospheric scale without scaling. The velocities from small-scale water-tank models are only about 20% of those of the air models. A simulation of water-tank experiments using computational fluid dynamics (Kristóf et al. 2009) agreed well with the corresponding water-tank experiments (Cenedese and Monti 2003).

3. A new velocity scale that considers the influence of the Prandtl number

Given the relatively small size and temperature excess of the urban area, the Boussinesq approximation is used and the Coriolis force is ignored. If one assumes that the urban area is circular with a uniform heat flux, the governing equations of the asymmetrical urban-dome flow can be written in a cylindrical coordinate system as follows:
e4
e5
e6
e7
where t is time; u and w are the radial velocity and vertical velocity, respectively; r and z are the radial and vertical coordinates, respectively; ρ0 and θ0 are the reference density and reference potential temperature, respectively; p and θ are the pressure excess and potential temperature excess, respectively; ν is the kinematic viscosity; κ is the thermal diffusivity; and g is again the gravitational acceleration.

By following the method of Lu et al. (1997a), the nondimensional form of the governing equations is obtained to analyze the nondimensional parameters used in the scaled models. The method used to define the length and time scales is the same as that used by Lu et al. (1997a), except in the definition of the radial velocity scale. The dependence of the scaling parameters upon the Prandtl number Pr has been discussed by Catalano et al. (2012b) for a case of penetrative free convection. The microscale velocity proposed by Catalano et al. (2012b) is wm = (g/N)(GaPr)−0.7(νN2f/g2)0.11 in the CBL, where Ga is the Gay-Lussac number, f is the time frequency (1/24 h on a daily basis), and the other variables have already been defined. Verzicco and Camussi (1999) gave the result that Re ≈ Pr−0.94 in a uniform heated enclosure. Hogg and Ahlers (2013) obtained the result that Re ≈ Pr−0.62 in a large-aspect-ratio enclosure with uniform heat flux. Here, we introduce nonhomogeneous surface heating.

The nondimensional parameters of interest, which can be defined on the basis of the scaling parameters shown in Table 2, are shown in Table 3. By substituting these nondimensional parameters into Eqs. (4)(7), the nondimensional governing equations can be obtained:
e8
e9
e10
e11
where now Fr = Ur/(ND) is the Froude number, Re = UrD/ν is the Reynolds number, and Pr = ν/κ is the Prandtl number.
Table 2.

Scales of different parameters: D is the urban diameter, and N = (gβ∂θa/∂z)1/2 is the background buoyancy frequency, where θa is the ambient potential temperature.

Table 2.
Table 3.

Nondimensional parameters.

Table 3.
The velocity scale can be obtained with the bulk model with the continuity, momentum, and energy equations, as in Lu et al. (1997b). The bulk model is illustrated in Fig. 3 for the water-tank model (Pr > 1). The mass of the inflow should equal the mass of the upward flow out of the control volume, as based on the mass conservation:
e12
where ρin and ρout are the density of the fluid flow into and out of the control volume, respectively; is the average velocity flow into the control volume; zr is the reverse height; D is the urban diameter; and and δυ are the average velocity flow out of the control volume and the plume hydrodynamic diameter (i.e., the diameter at which the mean vertical velocity reaches a certain percentage of the mean velocity at the plume center line), respectively. In a similar way, the plume thermal diameter δt (i.e., the diameter at which the mean temperature reaches a certain percentage of the mean temperature at the plume center line) is defined. Different parameters are defined in Fig. 3.
Fig. 3.
Fig. 3.

An illustration of the bulk model of urban-dome flow in a water-tank model (Pr > 1): Pr is Prandtl number; zi and zr are the mixed height and radial velocity reverse height, respectively; is the average velocity of inflow, also regarded as the velocity scale in the urban-dome study; is the average vertical velocity relating to the thermal boundary layer; is the average vertical velocity relating to the plume’s hydrodynamic diameter; δυ and δt are the plume hydrodynamic diameter and plume thermal diameter, respectively.

Citation: Journal of Applied Meteorology and Climatology 55, 10; 10.1175/JAMC-D-16-0099.1

Cenedese and Monti (2003) measured the average contraction ratio, defined as the ratio of the plume neck (the minimum urban-scale plume diameter) to the urban diameter, to be approximately 0.25. Husar and Sparrow (1968), Lu et al. (1997b), and Colomer et al. (1999) gave values for the contraction ratio that were in the range of 0.2–0.6. Colomer et al. (1999) also measured the distance between the neck position and the urban ground. The ratio of the distance to the urban diameter has an average value of 0.28 ± 0.13. In a similar way, the nondimensional ratio of the plume hydrodynamic diameter at the control volume outlet to the urban diameter is defined as Rυ = δυ/D. The nondimensional ratio of the plume thermal diameter at the control volume outlet to the urban diameter is defined as Rt = δt/D. By substituting Rυ and applying the Boussinesq approximation to Eq. (12), one obtains
e13
If the friction on the ground is ignored, the momentum equation can be given as
e14
where ΔTm is the average temperature difference between the mixed layer and the ambient air from the ground to zr. The kinetic energy at the background rural area is ignored.
The energy conservation equation can be written as
e15
where H0 is the surface heat flux and cp is the specific heat capacity. The physical meaning of is the average vertical velocity in the plume thermal diameter. For the three Eqs. (13), (14), and (15), there are four unknown parameters. The relationship between and must be found to obtain the velocity scale.
As illustrated in Fig. 3, and can be calculated as
e16
e17
The relationship between and can be deduced by using Eqs. (16) and (17) and Fig. 3 to obtain
e18
δυ and δt are again the plume hydrodynamic diameter and plume thermal diameter, respectively.
Worster (1986) solved plume-governing equations and obtained an asymptotic solution for large Prandtl numbers and numerical solutions for Prandtl numbers ranging from 1 to 10. The ratio of the hydrodynamic boundary layer thickness to the thermal boundary layer thickness changes with the Prandtl number following a power law, as shown in Fig. 4 and based on the data in Worster (1986). For a vertical heated plate, Bejan (2004) also derived a power-law solution δυ/δt = 1.095Pr1/2. This means that the ratio of to may be written as , where a and p are unknown positive numbers. The constants a and p cannot be determined here because the exact function f in Eq. (18) cannot be determined. By multiplying Eqs. (13)(15), we obtain
eq2
By substituting Eq. (18), we obtain
eq3
We thus obtain the average radial velocity flow into the control volume as
e19
where A is a constant that is independent of fluid properties and x is the exponent. The value of constant x in Eq. (19) cannot be analytically obtained because the constants a and p cannot be determined. Thus, given the influence of the Prandtl number, the radial velocity scale can be assumed to be in the form of Ur = [gβDH0/(ρcp)]1/3Prx.
Fig. 4.
Fig. 4.

The relationship between lυ/lt and Pr: lυ and lt are the hydrodynamic boundary layer thickness and thermal boundary layer thickness, respectively. The curve is fitted with the power function lυ/lt = CPrm, where C (=1.43) and m (=0.38) are positive constants (correlation coefficient squared R2 = 0.9981; the residual sum of squares is 0.003 04). The results are taken from Worster (1986) to demonstrate that the ratio of the hydrodynamic boundary layer thickness to the thermal boundary layer thickness changes with Prandtl number according to a power law.

Citation: Journal of Applied Meteorology and Climatology 55, 10; 10.1175/JAMC-D-16-0099.1

The exponent x can be determined by least squares fit using the available experimental data as follows:
e20
where Prw and Pra are the water Prandtl number and air Prandtl number, respectively; is the nondimensional radial velocity obtained in the water-tank experiments of Fig. 5 as shown by the open squares; and is the nondimensional radial velocity in the real-scale atmospheric models of Fig. 5 as shown by the filled circles.
Fig. 5.
Fig. 5.

A comparison of different nondimensional radial velocity profiles considering the influence of Pr. Filled circles represent the mean nondimensional radial velocity of real-scale models of the atmosphere in Fig. 2. Open squares represent the mean nondimensional radial velocity of water-tank experiments in Fig. 2. Filled blue triangles represent the nondimensional radial velocity in water-tank experiments corrected using Eq. (21) considering the influence of the Prandtl number (i.e., the new convection velocity scale); uD is the traditional velocity scale without consideration of the Prandtl number. Water corrected by UE, UG, and UL represents the corrected nondimensional radial velocity of water-tank experiments as based on the velocity scales of Eckert and Jackson (1950), Gebhart et al. (1988), and Ostrach (1982), respectively.

Citation: Journal of Applied Meteorology and Climatology 55, 10; 10.1175/JAMC-D-16-0099.1

The exponent x = 0.63 is obtained after the data in Table 4 (average Prw/Pra = 8.75) and Fig. 5 ( and ) were used in Eq. (20). The new radial velocity scale is thus defined as Ur = [gβDH0/(ρcp)]1/3Pr−0.63. If the turbulent Prandtl number (normally equal to 1) in the atmosphere is used, where average Prw/Pra = 6.3 in Table 4, the exponent x = 0.74 is obtained from Eq. (20).

Table 4.

Correction parameters used in Fig. 5.

Table 4.

To compare different experimental data obtained from air and water models, all data are scaled using the traditional velocity scale uD = [gβDH0/(ρcp)]1/3. The experimental data from water-tank models were corrected by using the following equation to account for the effect of the Prandtl number:
e21
Since the vertical velocity scale is defined as wD = uDFr, where uD is the radial velocity scale, the correction factor for the nondimensional vertical velocity at the center of the urban area can be obtained as
e22
where Fra and Frw are the Froude number in the air and water-tank experiments, respectively, and and are the nondimensional vertical velocity in the air and water-tank experiments, respectively.
It is interesting that the following relationship between the Reynolds number, the Rayleigh number, and the Prandtl number can be deduced from the new velocity scale with the exponent x = 0.63:
e23
where ReD = UrD/ν is the Reynolds number as based on the urban area diameter and RaD = gβD4H0/(νκ2ρcp) is the Rayleigh number as based on the surface heat flux and the urban diameter.
The Prandtl influence on the natural convection velocity scale with uniform heating on a horizontal surface was also discussed by Ostrach (1982), who suggested that the velocity scale should be UL = (ΔTL/Pr)1/2 when the hydraulic boundary layer and thermal boundary layer develop, where L is a length scale and ΔT is the temperature difference between the horizontal surface and the ambient fluid. If ΔTH0/(ρcpW), LziUL/N, and W are substituted into the velocity scale UL, UL = [gβDH0/(ρcp)]1/3Pr−1/3 can be obtained, where H0 is the surface heat flux, D is the urban diameter, N is the background buoyancy frequency, and W is the vertical velocity scale. Gebhart et al. (1988) also gave the vertical velocity scale along a vertical heated plate by an integration method, which is
eq4
and Eckert and Jackson (1950) gave a similar velocity scale by using an integral method with different exponents and constants, which is
eq5
where GrL is the Grashof number as based on the length scale L. The difference between the UG and UE velocity scales is due to the different assumptions on the velocity profiles. If ΔTH0/(ρcpW), LziUG/N, and W are substituted into UG and ΔTH0/(ρcpW), LziUE/N, and W are substituted into UE, the new velocity scales
eq6
eq7
can be obtained for the velocity scales of Gebhart et al. (1988) and Eckert and Jackson (1950), respectively. Each of the velocity scales mentioned above can be written in the form of U = B[gβDH0/(ρcp)]1/3f(Pr), where B is a constant and f(Pr) is a function of the Prandtl number, which suggests the influence of the Prandtl number on the velocity scale. The nondimensional radial velocity correction for water-tank experiments according to the velocity scales of Ostrach (1982) UL, Gebhart et al. (1988) UG, and Eckert and Jackson (1950) UE are also shown in Fig. 5. These existing convective velocity scales unfortunately do not work as well as expected. The urban-dome flows are not similar to those along a vertical or horizontal heated plate.

4. Results

The radial velocity correction for the results of the water-tank experiments is shown in Fig. 5. The values of the correction parameters are summarized in Table 4. The data for the real-scale models of the atmosphere and for water-tank models are the average values extracted from Fig. 2. The temperature in the water-tank experiments is usually in the range from 294 to 300 K, and therefore the Prandtl number for water is in the range from 5.8 to 6.8. The Prandtl number for air does not show an obvious change in this temperature range, and so 0.72 is used. The average of Prw/Pra, which is 8.75, is adopted in Eq. (20).

As seen in Fig. 5, the nondimensional velocities scaled with traditional velocity scales in water-tank experiments and in real-scale models of the atmosphere show a significant difference, which indicates that other factors should be included in the velocity scale. After correction with Eq. (21) considering the influence of the Prandtl number, the nondimensional radial velocity profile in the water-tank experiments agrees well with that of real-scale models of the atmosphere.

The correction of the nondimensional vertical velocity, the standard deviations of the horizontal velocity component, and the standard deviations of the vertical velocity component are shown in Figs. 68, respectively. According to Eq. (21), the vertical velocity scale is a function of both the Prandtl number and Fr. The Froude number varies for the different cases, and therefore one set of data [Catalano et al. (2012a) case Exp4, with Fr = 0.121, which is represented by open stars, and Catalano et al. (2012a) case Sim4, with Fr = 0.06, which is represented by inverted filled triangles] has been used to demonstrate the correction expressed by Eq. (22). The filled blue triangles represent the corrected data for the water-tank experiments. As can be seen in Fig. 6, the corrected value agrees well with the data of Catalano et al. (2012a) case Sim4. The Sim5 data (Catalano et al. 2012a) clearly differ from the simulation results of other nondimensional vertical velocities in air and have values similar to those in the water-tank experiments. This is caused by the small Froude number (0.02) in Sim5, which leads to a correction factor on the order of 1.

Fig. 6.
Fig. 6.

Nondimensional vertical velocity at the center of the heated area changing with a nondimensional height: (a) original data from water-tank experiments and air models, and (b) selected data for comparison with data from air models and water-tank experiments with Pr correction. See the text for a detailed explanation.

Citation: Journal of Applied Meteorology and Climatology 55, 10; 10.1175/JAMC-D-16-0099.1

The standard deviations of the velocity can be influenced by the stability of the background condition, and therefore one set of data [Lu et al. (1997b), with Fr = 0.077, which is represented by left-pointing open triangles, and Catalano et al. (2012a) case Sim3, with Fr = 0.09, which is represented by filled circles in Figs. 7 and 8] with similar Froude numbers has been chosen for comparison when the water-tank data have been corrected by the Prandtl number. The corrected data (filled blue triangles) from the water-tank experiments of Lu et al. (1997b) agree well with the data of Catalano et al. (2012a) case Sim3. The standard deviation of the velocity horizontal component and vertical component can also be influenced by the local buoyancy frequency, which is a damping parameter for the turbulence. The relatively hot fluid flows out of the urban area at a higher level, and the relatively cold fluid flows into the urban area at a lower level; thus a stronger stable stratification would be established at the urban edge, which is also proposed by Wang (2009). Wang (2009) suggested that the mixed height at urban edge is about two-thirds of the mixed height at urban center. The mixed height is slightly smaller than that at the rural area during daytime because of the stabilization at the urban edge. As shown in Figs. 58, the nondimensional parameters in the water-tank experiments show reasonable agreement with the results in air after correction considering the influence of the Prandtl number.

Fig. 7.
Fig. 7.

As in Fig. 6, but for nondimensional standard deviations of the horizontal velocity components.

Citation: Journal of Applied Meteorology and Climatology 55, 10; 10.1175/JAMC-D-16-0099.1

Fig. 8.
Fig. 8.

As in Fig. 6, but for nondimensional standard deviations of vertical velocity components.

Citation: Journal of Applied Meteorology and Climatology 55, 10; 10.1175/JAMC-D-16-0099.1

The laminar characteristics of area-source plumes have been reported by other researchers. Colomer et al. (1999) and Fannelöp (1994) referred to the flow in near-source convergence regions as “laminar plumes” because entrainment does not occur in this region and because the flow is stabilized by acceleration. The flow above the neck is unstable and turbulent, and large-scale eddies cause entrainment. Field measurements carried out by Hu et al. (2013) demonstrated that the friction velocities and turbulent kinetic energy decreased to negligible levels in the urban-dome-flow region in a strong UHI. A similar phenomenon occurs in plumes generated by fire. Heikes et al. (1990) indicated that large fire plumes (i.e., those with a radius greater than 5 km) are fundamentally different from small source plumes. The entrainment of ambient air is significantly less in the larger plumes than in smaller plumes. The introduction of the Prandtl number makes it possible to take into account the effect of nonturbulent or less-turbulent regions on the scaling parameters.

5. Discussion

Different scaling parameters are used for different physical problems. The scaling parameters related to the urban-dome flow that have been proposed by different researchers are summarized in Table 5.

Table 5.

Different scaling parameters related to urban-dome flow: D and zi represent the urban diameter and mixed height, respectively; z is the local coordinate and represents the distance between the study position and the source; h is the thermal rise height; is the buoyancy flux; ρ0 is the reference density; and are the time-averaged kinematic turbulent heat flux and density flux, respectively, at the surface; H0 is the surface heat flux (W m−2); and Q is the kinematic heat flux (m K s−1).

Table 5.

The ratio of the horizontal length scale to the vertical length scale is one of the key parameters for evaluation of which length scales should be used in the scaling models. In general, if the aspect ratio of the flow field is very large, the mixed height would be used as the length scale, such as in the CBL and in Rayleigh–Bénard convection. If the aspect ratio is sufficiently small for the self-similar region to be retained when the height is greater than a certain value, the local height z would be used as the length scale, and the plumes can be regarded as point-source plumes in the self-similar region. Both the source diameter D and local height z would be used as length scales in the near-source region for this kind of flow, however, because both D and z can influence the flow. The surface characteristics (roughness, height, frontal-area density, and building-area density) can be also important geometrical parameters in the near-ground region. The aspect ratio of the urban-dome flow lies between the above two situations. (With an urban diameter on the order of 10–40 km and a mixed height on the order of 0.2–1.5 km, the aspect ratio of the urban-dome flow is on the order of 6–200.) Lu et al. (1997b) noted that the geometry of urban-dome flows differs significantly from that of typical plumes. They also pointed out that there is little entrainment from the environment and that the similarity region does not exist in the urban-dome flow. The urban diameter D was chosen as the length scale in the theory of Lu et al. Hidalgo et al. (2010) suggested that the mixed height should be used as the length scale during UHI-induced breezes during the day, because the CBL would be sufficiently strong to dominate the flow during the day.

If the size of the urban area is sufficiently large and the heat flux is relatively uniform, the urban-dome flow at night may also form multiple cells. The mixed height over an urban area would then be used as the length scale instead of the urban size. The size of an aspect ratio that would result in multiple cells in the urban-dome flow forming at night remains unknown. The aspect ratio of individual cells in the CBL is on the order of 20–30, and so the aspect ratio of the urban-dome flow should at least be on the order of several hundreds to contain multiple cells. The exact value of the borderline aspect ratio that would lead to multiple convective cells over an urban area is worth exploration to gain a better understanding of the urban-dome flow and related scaling methods.

6. Conclusions

Small-scale water models have been a very useful tool in the exploration of the mechanisms by which urban domes and their associated wind flows are formed. The scaling parameters used for the CBL, urban-dome flow, UHI-induced breezes during the day, and area-source thermal plumes are reviewed in this paper. The scaling characteristics were categorized according to different aspect ratios. For convective flow with a large aspect ratio (approximately 1000), such as the CBL, the mixed height zi was used as the length scale for calculation of the velocity scale. For convective flow with a small aspect ratio (0.01–0.1), such as area-source plumes, the local height z was used as the length scale distant from the source region and the combination of the local height z and the source diameter D was used as the length scale in the near-source region. For convective flow with a medium aspect ratio (from several to hundreds), such as UHIC, the source diameter D was adopted as the length scale.

The Prandtl number of the fluids used in scaled models was found to influence the scaling of urban-dome flows. A new velocity scale that considers the influence of the Prandtl number is proposed to account for the effects of nonturbulent or less-turbulent regions in urban-dome flows and was found to work very well for both mean velocities and turbulence fluctuations through comparisons with water-tank models, simulations using computational fluid dynamics, and field measurement data. This new development is expected to enable small-scale models to be more useful in urban wind studies. The new convective velocity scale could be extended to water-modeling studies of other dominant buoyancy-driven airflows.

Acknowledgments

This work is supported financially by a Research Grants Council Collaborative Research Fund project (HKU9/CRF/12G) of the Hong Kong SAR government.

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1

For consistency, the aspect ratios used in this paper are defined as the ratio of the horizontal length scale to the vertical length scale, which differs from the definition used in plume theory (in which it is defined as the ratio of the height to the width of the plume).

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