Macroscopic simulations of turbulent flows through high-rise building arrays using a porous turbulence model

Building and Environment 49 (2012) 41e54

Contents lists available at SciVerse ScienceDirect

Building and Environment journal homepage: www.elsevier.com/locate/buildenv

Macroscopic simulations of turbulent flows through high-rise building arrays using a porous turbulence model Jian Hanga, b, Yuguo Lia, * a b

Department of Mechanical Engineering, The University of Hong Kong, Haking Wong Building, Pokfulam Road, Hong Kong SAR, China Guangdong Provincial Key Laboratory of Building Energy Efficiency and Application Technologies, Guangzhou University, Guangzhou, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 22 July 2011 Received in revised form 12 September 2011 Accepted 13 September 2011

Wind flowing through urban areas may help diluting pollutants in street networks. General microscopic numerical techniques have difficulty in simulating wind through city-scale urban areas with thousands of buildings because the required computational load is difficult to afford. We considered urban canopy layers with buildings and street networks as porous media and used a porous turbulence model to macroscopically study urban airflows. High-rise porous building arrays with uniform building heights or building height variations were studied (building height/street width, i.e. H/W ¼ 2 or 2.67; the porosity or the fraction of air volume in urban areas is 4 ¼ 0.75). A single domain approach was used to account for the interface conditions. Microscopic simulations using RANS k-3 turbulence model and validated by wind tunnel data were also carried out to model the form drag produced by buildings and calculate spatially-averaged flow quantities to estimate macroscopic simulation results using the porous turbulence model. Results showed that, with a parallel approaching wind, the present porous turbulence model may predict macroscopic mean flows through porous building array generally well if suitable porous parameters are modelled, meanwhile, some microscopic flow information is lost but the computational requirements are effectively reduced. With a power-law approaching wind, a taller porous building array may experience greater macroscopic velocity if the length of porous region is effectively limited. Further investigations are still required to evaluate macroscopic turbulence predictions and apply present porous turbulence model for real urban areas or cities with various wind directions. Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved.

Keywords: Building array High-rise Porous turbulence model Numerical simulation Spatially averaging

1. Introduction In high-rise compact urban areas like Hong Kong, Manhattan in New York and Tokyo, the urban canopy layer (UCL) where most urban residents work and live, consists of thousands of solid highrise buildings and relatively narrow street networks between them. Such urban air environments usually differ from those in rural areas due to high-rise urban structures and crowded human activities, i.e. produce an “urban pollutant island” [1] and “urban heat island” [2] in comparison to their surroundings. Wind from rural areas may help diluting the contaminated or heated air in urban areas. However, rural wind through high-rise compact urban areas may be weakened quickly because high-rise buildings produce strong resistances to the approaching wind and the ventilation flow rates through relatively narrow street networks (i.e. the porosity is small) is limited. Investigations on the mechanism of wind flowing * Corresponding author. Tel.: þ852 2859 2625; fax: þ852 2858 5415. E-mail address: [email protected] (Y. Li).

through such high-rise urban areas can help reducing urban air pollution and improving city ventilation. Britter and Hanna [3] classified studies of the flow and dispersion in urban areas into four scales, i.e. the regional scale (up to 100 or 200 km), the city scale (up to 10 or 20 km), the neighbourhood scale (up to 1 or 2 km) and the street scale (less than 100e200 m). Experiments and Computational Fluid Dynamics (CFD) simulations using large eddy simulations (LES) or Reynolds-averaged NaviereStokes (RANS) turbulence models have been widely carried out to study urban airflows in two-dimensional (2D) street canyons [4e6] and three-dimensional (3D) urban areas [7e11] from street scales to neighbourhood scales. However CFD simulations of city-scale urban airflows are rare so far because the requirement of the computational power is difficult to afford using present computer techniques. For example, even performing CFD simulations with the less expensive RANS turbulence models, at least 10 cells per building edge are required and the minimum grid resolution should be about 10% of the building scale [9], so a city-scale urban area with thousands of buildings requires billions of grids. We considered an urban area as

0360-1323/$ e see front matter Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2011.09.013

42

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

porous media (see Fig. 1a), then macroscopic airflows through porous urban canopy layers are similar with airflows through agricultural canopies [12,13] or other porous media. In urban-type porous media, the flow information between buildings is spatially averaged by the volume averaging technique and the effects of buildings to urban airflows are modelled by porous turbulence models as additional source or sink terms in the governing equations for momentum and turbulence transportation. In ‘continuously’ distributed porous city regions, buildings disappear and the minimum grid number per

building edge required in microscopic simulations [9] does not exist in macroscopic simulations using porous turbulence model, therefore, the grid number can be reduced effectively, especially in porous regions with small gradients of flow variables the grid size can be larger than the building scale (or size of solid objects in porous media). All porous turbulence models so far were derived from the volume-averaging technique on a Representative Elementary Volume (REV) and based on Darcy’s Law or its extension. These models focus on the macroscopic flow characteristic in the REV and

Fig. 1. (a) Analogy of an urban canopy layer and a porous city region. Model descriptions in wind tunnel tests: (b) Building arrays with uniform heights in Case [2-2, 6] or Case [2.672.67, 6], (c) an array with a building height variation in Case [2-2.67, 9]. Each case name denotes an aligned square array [H1/BeH2/B, row number] (Street width W ¼ building width B, the porosity is 0.75).

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

neglect detailed flow information inside the REV. All the volumeaveraging techniques on microscopic equations can generate several additional terms to model the viscous and form drag forces in porous media [14e19]. When the prevailing wind dominates the flow through urban areas, thermal effects can be disregarded and urban airflows can be considered as isothermal high-Reynoldsnumber turbulent flows. Antohe and Lage [14] mathematically developed a good two-equation porous turbulence model for such flows in which the closure coefficients are the same as those in RANS standard k-3 turbulence models. Hang and Li [20] treated urban-like cube arrays (the aspect ratio i.e. building height/street width H/W ¼ 1, the porosity f ¼ 0.75) as porous media and solved macroscopic urban airflows using the model developed by Antohe and Lage [14]. The turbulent airflows between individual obstacle elements are unresolved, and the effects of the buildings on urban airflows are modelled as sink or source terms for the momentum and turbulence transportations. Hang and Li [20] confirmed that this porous turbulence model may predict macroscopic mean flows in porous cube arrays generally well. Besides Hang and Li [20], Lien and Yee [21] also used a modified k-3 turbulence model [18] that they derived previously to study macroscopic airflows through and over a 7-row cube array, but didn’t evaluate macroscopic simulation results. In addition, to theoretically study the macroscopic wind conditions through the urban canopy layers, some canopy approaches were developed. Belcher et al. [22] revised a model for vegetation canopy flows [13] to investigate the macroscopic adjustment and wind reduction through sparse cube arrays (H/W ¼ 0.5, f ¼ 0.89) in which the buildings act as bodies producing a point drag force and the dispersive stress can be neglected because the gap between obstacles is large. Then Conceal and Belcher [23] extended this model to inhomogeneous canopies including variable canopy densities and variable canopy heights. Their models were considered invalid for high-rise urban canopy layers in which the skimming flow regime exists. Moreover, MacDonald [24] and Di Sabatino et al. [25] theoretically predicted the local spatially-averaged (macroscopic) velocity profile in cube building arrays based on the assumption that a balance exists between the form drag and the local shear stress under statistics information of all buildings. These models [24,25] are not suitable to study macroscopic wind conditions as rural wind flows through and above 3D high-rise urban areas in which both windward and leeward boundaries of urban canopy layers exist and the form drag is not in a balance with the shear stress. Overall, this paper aims to quantify the macroscopic (spatiallyaveraged) wind conditions through some high-rise porous urban areas. Since Hang and Li [20] confirmed that the porous turbulence model developed by Antohe and Lage [14] is promising to study low-rise porous cube arrays with a uniform height (H ¼ B ¼ W, f ¼ 0.75), this paper applied and extended the same approach on some high-rise porous building arrays (H/W  2, f ¼ 0.75) with uniform building heights or with a building height variation. In addition, the microscopic flow field in such high-rise building arrays has been experimentally and numerically investigated in previous literatures [10,11], which may help understanding the form drag produced by buildings and provide evaluations for macroscopic simulation results as well as clarify the difference between two simulations.

2. Methodology and model descriptions 2.1. Microscopic and macroscopic transport equations We regarded airflows through high-rise building arrays as a kind of incompressible, isothermal turbulent flow through and over

43

porous media. Both microscopic simulations and macroscopic simulations using a porous turbulence model were carried out. There are two kinds of average operators, the time-average operator and local volume-average operator on NaviereStokes equations. By the time-average operator, the general fluid property j can be divided into the time average j and the time fluctuating component j0 (see Eq. (1)).

1 j¼ Dt

tþ Z Dt

j ¼ j þ j0

jdt;

(1)

t

The well-known RANS turbulence models including RANS k-3 turbulence models and Reynolds stress model (RSM) are commonly used to close the time-averaged NaviereStokes equations. We used the standard k-3 turbulence model to microscopically simulate the time-averaged mean flows and turbulence characteristics in urban canopy layers when the grid number is limited and the calculation load is affordable. All microscopic time-averaged governing equations are as follows:

  v rui ¼ 0 vxi   v rui uj vxj   v ruj k vxj   v ruj 3 vxj

(2)

v ¼ vxj v ¼ vxj

"

m mþ t sk

"

m mþ t sk

"

v ¼ vxj

m mþ t s3







vui vxj

vk vxj v3 vxj

#

 



vp 2 vk þ r vxi 3 vxi

(3)

# þ Gk  r3

(4)

# 3

3

2

þ c3 1 Gk  rc3 2 k k

 C3 1 ; C3 2 ; Cm ; sk ; s3 ¼ f1:44; 1:92; 0:09; 1:0; 1:3g

(5)

(6)

wheremt ¼ rCm is the turbulent viscosity, m is the viscosity, Gk is the turbulence kinetic energy production ð ¼ mt ðvui =vxj Þ ððvui =vxj Þ þ ðvuj =vxi ÞÞÞ. The flow quantities (ui , p, k, 3 ) are timeaveraged velocity components, pressure, turbulence kinetic energy and its dissipation rate. By using the volume-average operator, the property j can be divided into the intrinsic average < j >f and its spatial variation ji within the Representative Elementary Volume (REV) in porous media. The macroscopic time-averaged quantities (i.e. the volume average of time-averaged variables) for flows through porous media can be acquired by volume averaging the corresponding microscopic time-averaged quantities over a volume of REV (DV). k2 =3

< j>f ¼

< j>v ¼

1 DVf 1

DV

Z

jdV

(7a)

DVf

Z DVf

jdV ; < j>v ¼ f < j>f ; f ¼

DVf DV

(7b)

where DVf is the fluid volume in a special volume (REV)DV, f is the fraction of fluid volume in porous media, i.e. the porosity. < j >v and < j >f are the volumetric and intrinsic averages of a timeaveraged quantity j which are related by the porosity f. To simplify the equations, we wrote macroscopic time-averaged variables < j >f (i.e. the spatial intrinsic averages of microscopic time-averaged variables) into jf in the following governing equations.

44

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

Antohe and Lage [14] derived a macroscopic k-3 model for macroscopic time-averaged transport equations in high-Reynoldsnumber incompressible turbulent flows in porous media:

  v rfufi

¼ 0

vxi   f f v rfui uj

(8)

" # f vfui v ðmJ þ mt Þ  ¼ vxj vxj

vxj

vfpf 2 vfkf þ r 3 vxi vxi

!

uForch

udarcy zfflfflfflfflfflffl}|fflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ m rC f fufi f pffiffiffiFffi fQ f fufi K K

  v rfufj kf vxj

v ¼ vxj

"



m vfkf mJ þ t sk vxj

(9)

#

TKEForch TKEdarcy TKEgenzfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{

TKE Fk

zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ zfflffl}|fflffl{ m f 2 CF 8 f f C f  rf3 þfGk 2f fk f pffiffiffiffi rQ fk þ 2ff2 pFffiffiffiffi Fk K K3 K (10)   f v rfuj 3 f vxj

v ¼ vxj

"



m vf3 f mJ þ t sk vxj

#

EDForch1

þ c3 1 f

3

f

kf

Gk  J rc3 2 f

3

f

kf

3

f

EDForch2

zfflfflfflfflfflffl}|fflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ m 8 C 8m C vkf vQ f 2f f3 f  f3 pFffiffiffiffi rQ f 3 f  f3 pFffiffiffiffi K 3 3 K K vxr vxr EDForch3 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl }|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ 2 0 1 !3 f f f f f 2 f 2 u u u u v u vu C v @ j iA j i v i i 5 þ2f2 f pFffiffiffiffi 4mnt þ 2mnt f vxr vxr vxj Qf Q vx2r vxj K EDdarcy

(11) f

Fk ¼ mt

f

0 1 f f f vuj vui vui @ A ; Gk ¼ mt þ vxi vxj vxj vxi f

uj ui vuj Qf

(12)

where r is the fluid density, ufi ; pf ; kf ; 3 f are intrinsic averages of time-averaged variables (the velocity components, pressure, turbulent kinetic energy and its dissipation rate). m is the viscosity

and mt ðmt ¼ Cm rkf 2 =3 f Þ is the turbulent viscosity in porous media, ~ =m is the viscosity ratio which can be assumed to be equal to J ¼ m one for most applications in porous media [14], K is the permeqffiffiffiffiffiffiffiffiffiffi ability, CF is the Forchheimer coefficient. Q f ¼ ufi ufi is the intrinsic averages of velocity. The closure coefficients in Eqs. (8)e(12) are the same as those in Eq. (6). In the right side of Eqs. (9) and (10), the Darcy terms (i.e. udarcy and TKEdarcy) and the Forchheimer terms (i.e. uForch and TKEForch) are used to model microscopic viscous drag generated by the fluid itself and microscopic form drag produced by solid objects on the transportation of momentum and turbulent kinetic energy (TKE) in porous media. All these terms are always negative and act as sink terms. It represents that they strengthen the reduction of momentum and turbulence transportation in porous media. However in Eq. (10), the second part (i.e. TKE_Fk) of the Forchheimer terms for TKE may act as a sink or source term for turbulence transportation because it may be either negative or positive in porous media depending on the velocity components

and their derivatives. Similarly in Eq. (11), the Darcy term (i.e. EDdarcy) and the first part (i.e. EDForch1) of the Forchheimer terms are also negative which result in a decrease in the dissipation rate of TKE, i.e. weakening the depletion of TKE. Hang and Li [20] analyzed the effects of the above terms in porous cube arrays. They found that, the Forchheimer terms (uForch, TKEForch and EDForch1) are always much greater (in order of 100 times) than the Darcy terms (udarcy, TKEdarcy and EDdarcy) in Eqs. (9)e(11) because the viscosity (m) is small in fully-developed turbulent urban airflows with high Reynolds numbers. In addition, we neglected the effect of EDFroch2 and EDForch3 since they are about 100 and 1000 times smaller than EDForch1 due to the similar reason (i.e. the viscosity (m) is small) according to Hang and Li [20]. The macroscopic urban airflow is a kind of turbulent flow through and over porous media with some porosity and permeability. As shown in Fig. 1a, the whole domain consists of the porous city region (i.e. the urban canopy layer) and a clear fluid region (i.e. non-porous fluid region) with a macroscopic interface. The mathematical treatments of the interface conditions are important to the mass, momentum and turbulence transfer between two regions. Since Eqs. (8e12) for porous regions may become the same as those in the clear fluid region (see Eqs. (2e6)) if the porosity f is 1, the permeability K is infinite, and the Forchheimer coefficient CF is zero, so Hang and Li [20] used single domain approach to deal with the interface conditions, i.e. to use the same equations (Eqs. (8e12)) but different porous parameters (the porosity f(x, y, z), the permeability K(x, y, z) and the Forchheimer coefficient CF(x, y, z)) in the two regions. 2.2. Model descriptions in wind tunnel experiments We are interested in urban airflows in high-rise urban areas consisting of buildings of 60 m or 80 m tall (i.e. about 20 or 27 floors) at full scale. Wind tunnel building models were made in Hang et al. [11] with the scale ratio of 1:1000 between the experimental models and those at full scale, i.e. the experimental model height is 60 mm or 80 mm. As shown in Fig. 1b,c, the building width (B) and street width (W) are all constant (B ¼ 30 mm, W1 ¼ W2 ¼ W ¼ 30 mm, i.e. 30 m wide at full scale), so the building area density islp ¼ ðB  BÞ=ððB þ WÞðB þ WÞÞ ¼ 0:25 and the porosity is f ¼ 0.75. The building heights are H ¼ 60 mm ¼ 2B or H ¼ 80 mm ¼ 2.67B, so the street aspect ratios (H/W) are 2 or 2.67. For all aligned building arrays, the main streets and the secondary streets are parallel and perpendicular to the approaching wind respectively. We defined the buildings towards to the downstream regions as rows No. 1, 2, 3 . N (N is the final row number) and x, y, z are the stream-wise, lateral and vertical directions respectively. We named aligned square building arrays as Case [building heights H1/ BeH2/B, the total row number]. Therefore Case [2-2,6] and Case [2.67-2.67, 6] (see Fig. 1b) denote two 6-row aligned arrays with uniform building heights of H1/B ¼ H2/B ¼ 2 or 2.67 (the total street length L ¼ 11B, i.e. 330 m at full scale). Similarly Case [2-2.67, 9] (see Fig. 1c) represents the 9-row aligned array with a building height variation (for rows of No. 1, 3, 5, 7, 9, building height H1/B ¼ 2 and for rows of No. 2, 4, 6, 8, H2/B ¼ 2.67; L ¼ 17B, i.e. 510 m at full scale). As shown in Fig. 1b,c, the location of the windward boundary of building arrays is x/B ¼ 0. The plane of y/B ¼ 0 is the symmetric plane of the middle main street for building arrays. The plane of y/ B ¼ 1 is the symmetric plane of the secondary streets neighbour to the middle main street (y/B ¼ 0). In Hang et al. [11], the velocity and turbulence intensity were measured using hotwire anemometers including horizontal profiles along the street centreline of the middle main street (y/B ¼ 0) at a height of z ¼ B and vertical profiles at many points in y/B ¼ 1. In Fig. 1b, Points V1eV5 are the centre points inside the secondary streets of No. 1e5, for which the

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

locations are at y/B ¼ 1 and x/B ¼ 1.5e9.5. In Fig. 1c, Points V1eV8 are the centre points inside the secondary streets of No. 1e8, for which the locations are at y/B ¼ 1 and x/B ¼ 1.5e15.5. All measurements were carried out in an aerodynamics boundary layer wind tunnel (closed-circuit type) which is located at the Laboratory of Ventilation and Air Quality, University of Gävle, Sweden. More detailed information about wind tunnel tests can be found in Hang et al. [11]. 2.3. Numerical set-ups In microscopic CFD simulations, we used the same high-rise square building models as those in wind tunnel experiments performed by Hang et al. [11]. The building arrays in wind tunnel tests are sufficiently wide in the span-wise direction. Experimental data [11] confirmed that wind in the middle main street is mainly affected by the external flow above these building clusters, and is affected little by the external flows beyond the lateral boundaries of building arrays. To reduce the calculation time in CFD simulations, we only considered the middle column of building arrays and only used half of this column which is surrounded by dash lines in Fig. 1b,c. So the computational domain in the lateral (i.e. y) direction is 30 mm (B) wide. In the vertical (i.e. z) direction the computational domain is 750 mm (25B) high. In the stream-wise direction (i.e. x), the distance from the upstream domain inlet to the windward boundary of building arrays is 33.3B, and that from the

45

leeward boundary of building arrays to the downstream domain outlet is 121.3B. We used a no slip wall boundary conditions at all wall surfaces and a zero normal gradient boundary condition at the domain outlet, the domain top and two lateral (or span-wise) symmetrical boundaries. Vertical profiles of the velocity (U(z)) and turbulence intensity (I) were measured in the upstream free flow (see Fig. 2a) where the velocity U(z) actually equals the stream-wise velocity u (z) because the span-wise and vertical velocity are zero ðv ¼ w ¼ 0Þ in the free flow. U(z) in Fig. 2a is a kind of power-law profile U(z) z 2.9  (z/B)0.1616 as z < 6.7B and a linear profile U(z) z 3.91 m/s as z > 6.7B. Such vertical profiles were used to provide boundary conditions at the domain inlet, where the turbulent kinetic energy k and its dissipation rate 3 were 3=4 calculated by the measured data (k ¼ 1.5(IU)2, 3 ¼ Cm k3=2 =lt , Cm ¼ 0.09 and lt is the turbulent characteristic length scale). In microscopic CFD simulations, we not only studied building clusters of the same number of rows (i.e. 6 rows or 9 rows, 330 me510 m at full scale) as those in wind tunnel tests but also investigated two more test cases in a neighbourhood scale (L ¼ 35B, 1.05 km at full scale), i.e. Case [2.67-2.67, 18], and Case [2-2, 18]. In microscopic CFD simulations, Hang et al. [11] performed a detailed grid independence study on Case [2.67-2.67, 6] using three kinds of grids (see Table 1). They found that microscopic numerical results using the medium grid arrangements change little when the grids become finer. So we used the medium grid arrangements as the default choice for all test cases which were summarized in Table 2.

Fig. 2. (a) Vertical profiles of the velocity (U) and turbulence intensity (I) measured in the upstream free flow. (b) Gird arrangements of microscopic simulations in xez and xey in Case [2.67-2.67, 6]. Gird arrangements of macroscopic simulations using porous turbulence model in (c) Case [2.67-2.67, 6] (197(x)  87(z)) and (d) Case [2-2.67, 9] (388(x)  118(z)).

46

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

Table 1 Three kinds of grid arrangements in microscopic simulations of Case [2.67-2.67, 6]a. In CFD simulations

Grid number in x, y, z

Minimum dx in canyon (mm)

Minimum dy in canyon (mm)

Minimum dz in canyon (mm)

Fine grid Medium grid Coarse grid

528  39  121 361  29  100 225  21  87

0. 47 0.94 3

0.29 0.54 1

0.2 0.4 0.7

a

Each case name denotes an aligned square array [H1/BeH2/B, row number].

Fig. 2b shows an example of medium grid arrangements in Case [2.67-2.67, 6]. The grid size in the stream-wise direction (x) increases with a ratio of 1.1 from the windward boundary of building arrays to the domain inlet and from the leeward boundary of building arrays to the domain outlet. The grid size in the vertical direction (i.e. z) above buildings also increases with an expansion ratio of 1.1. The total grid number for all test cases is about 1 million to 2.5 million (see Table 2). We also analyzed the grid independence of spatially averaged (macroscopic) properties and the sectional form drag by buildings in Case [2.67-2.67, 6]. For macroscopic simulations using the porous turbulence model, we used the same computational domains as those in microscopic CFD simulations but with a different dimension. Considering the symmetrical characteristics, the 3D microscopic urban airflows through groups of buildings are replaced by 2D macroscopic airflows through continuous porous media. We performed two test cases using the porous turbulence model, i.e. Case [2.67-2.67, 6] with a uniform canopy height and Case [2-2.67, 9] with a canopy height variation. For Case [2.67-2.67, 6], the coarse grids (197(x)  87(z)) and the fine grids (279(x)  100(z)) were used finding that there is little difference for the simulation results using the above two grids, therefore the coarse grid of 197(x)  87(z) can be thought sufficient. For Case [2-2.67, 9], the grids of 388(x)  118(z) were used. The grid arrangements of these two cases were displayed in Fig. 2c,d. To carry out both microscopic and macroscopic simulations, we developed the ‘Ventair’ code. The transport equations (see Eqs. (8e12)) were expressed in time-implicit and conservative finite difference form on a staggered grid. The transport equations for the momentum and turbulent properties were discretized by finite volume techniques. The hybrid upwind/central differencing scheme was used to discretize the advection terms, with an option of second-order upwind scheme and QUICK scheme. The discretized differential equations were solved by the SIMPLE algorithm. We used an under-relaxation factor of 0.5 for the mean velocities and pressure, and values of less than 0.5 for the turbulent properties to avoid divergence problems.

2.4. Volume averaging technique and key parameters in porous turbulence model Because we mainly studied high-rise urban canopy layers in a neighbourhood scale (about 1 km) and a scale less than it, the grid number in microscopic CFD simulations by RANS k-3 turbulence model is only several millions using a technique in the selection of computational domain. So we also performed microscopic CFD simulations which were evaluated by wind tunnel data and were used to calculate spatially-averaged properties and model the form drag in high-rise building arrays (H/W ¼ 2 or 2.67; f ¼ 0.75). We regarded the urban canopy layer as spatially continuous porous media which acts as sinks/sources to the momentum and turbulence properties of urban airflows. We used Eq. (7a) to calculate macroscopic properties by volume averaging the corresponding microscopic time-averaged quantities over REV units. The filter length of each REV unit should be larger than scales in a microscopic sense and also should be small enough to preserve as much information of the unfiltered flow as possible in a REV unit. For flows in high-rise building arrays, a suitable representative elementary volume (REV) consists of a unit of building and the space surrounding it. Fig. 3a,b shows two methods for the definition of REV units (Method 1 and Method 2) in aligned arrays with uniform building heights. A building array with N rows consists of N REV units. According to Hang and Li [20], macroscopic quantities

a

Method 1 in REV definition

REV unit i-2

REV unit i

REV unit i-1 z

No i

No i-1

[2.67-2.67, 6] [2-2, 6] [2-2.67, 9] [2.67-2.67, 18] [2-2, 18]

b

Method 2 in REV definition

REV unit i

REV unit i-2

a

Porosity Aspect Row Definition Grid number ratio (H/W) number of REV in x, y, z (medium grid size) 0.75 0.75 0.75 0.75 0.75

2.67 2 2 or 2.67 2.67 2

6 6 9 18 18

Method Method Method Method Method

1 1 2 1 1

361 361 472 841 841

    

29 29 27 29 29

    

z

100 95 118 100 95

For Method 1, as shown in Fig. 3a, the starting and ending points of REV unit i are at windward edge of Building No. i and No. i þ 1, i.e. xi ¼ (i  1)(B þ W2), xiþ1 ¼ i(B þ W2). For Method 2 (see Fig. 3b), the starting and ending points of REV unit i are at the middle location of two neighbouring secondary streets, i.e. xi ¼ (i  1)(B þ W2)  0.5W2 and xiþ1 ¼ i(B þ W2)  0.5W2.

No i+1

Method 1 in REV definition

REV unit i-1

a

REV unit i+1

x

Table 2 Summary of test cases and medium grid arrangements for all microscopic CFD simulations. Case

REV unit i+2

REV unit i

No i-1

REV unit i+2

REV unit i No i

REV unit i+1 No i+1

x

c

Method 2 in REV definition

x dz Fig. 3. Methods for the definition of REV units in aligned arrays with uniform building heights: (a) Method 1 and (b) Method 2. (c) The definition of thin horizontal slabs at the height of z.

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

calculated by two methods of REV definitions in cube arrays are little different in most regions except those at the first unit (No. 1) and at the final unit (No. N). We chose Method 1 (see Fig. 3a) as the default method for arrays with uniform heights and Method 2 (see Fig. 3b) for arrays with building height variations in the following calculation. Fig. 3c shows the definition of thin horizontal slabs at the height level of z. Since half a column of the street (see Fig. 1b,c) was used, the volume of a thin horizontal slab is DV ¼ 0.5(B þ W1) (xiþ1  xi)dz, where i is the No. of the REV units (from 1 to N) and dz is the interval distance between two neighbour grids in the vertical direction. For Method 1, xi ¼ (i  1)(B þ W2), xiþ1 ¼ i(B þ W2). For Method 2, xi ¼ (i  1)(B þ W2)  0.5W2 and xiþ1 ¼ i(B þ W2)  0.5W2. The sectional spatially-averaged (macroscopic) properties can be calculated by volume averaging (see Eq. (7a)) the corresponding time-averaged quantities over thin horizontal slabs (see Fig. 3c).

47

Important parameters for porous media include the porosity (f), the permeability (K) and the Forchheimer coefficient (CF). The porosity (f) is defined in Eq. (7b) as the fraction of fluid volume in the total volume of a REV unit. The permeability (K) in the Darcy terms for the viscous drag and the Forchheimer coefficient (CF) for the form drag can be approximately modelled by the Ergun equation [26] as follows:

   K ¼ f3 d2p = 150 1  f2

(13a)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi CF ¼ 1:75= 150f3

(13b)

where dp is the equivalent spherical diameter of solid objects in porous media. Here dp ¼ 6 Vp/Ap where Vp is the volume of a building and Ap is its total surface area.

Fig. 4. (a) Vertical profiles of velocity at Point V2 in Case [2.67-2.67, 6] including wind tunnel data and numerical results using three kinds of grids. (bed) The grid independence of spatially-averaged (macroscopic) properties in horizontal thin slabs of six REV units in Case [2.67-2.67, 6] using three kinds of grid (eef) Vertical profiles of sectional Forchheimer coefficients using the fine grids (CF1 (z)) and medium grids (CF2 (z)).

48

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

Fig. 5. In Case [2.67-2.67, 6]: (a) 2D streamline and distribution of velocity in y ¼ B and z ¼ B, and (b) 2D streamline and distribution of stream-wise velocity ðuÞ and vertical velocity ðwÞ in y ¼ 0. (c) Macroscopic stream-wise velocity (uf) and vertical velocity (wf) in 2D porous regions.

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

The Forchheimer coefficient (CF) shows the effect of form drag produced by building units to urban airflows. It may vary with variation of vertical location (z) and in different REV units. To obtain better prediction of the form drag, the sectional Forchheimer coefficient CF(z) in horizontal thin slabs of each REV unit can be calculated by the pressure difference between the windward and leeward wall surfaces of each building using Eqs. (14) and (15).

CF ðzÞ ¼

pffiffiffiffi fx ðzÞ K rQ f ðzÞuf ðzÞ Z

fx ðzÞ ¼

pðzÞnx dS=DVðzÞ

(14)

(15)

S

where Qf(z) and uf(z) are local intrinsic averages of time-averaged velocity and the stream-wise velocity component inffi each thin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi horizontal slab of REV units ðQ f ¼ uf 2 þ vf 2 þ wf 2 Þ, K is the permeability, fx(z) is the sectional form (pressure) drag in each thin horizontal slab, DV(z) is the total volume of each thin horizontal slab. Overall, the microscopic CFD simulations were first carried out and validated by wind tunnel data, then macroscopic profiles on flow characteristics in urban areas were analyzed by volume averaging the microscopic time-averaged quantities in REV units and such profiles may provide evaluation for macroscopic simulation results using the porous turbulence model. In addition, sectional Forchheimer coefficient (CF(z)) in thin slabs of REV units can be calculated using Eqs. (14) and (15).

49

3. Results and discussion 3.1. Inporous building arrays with uniform heights (f ¼ 0.75) 3.1.1. Macroscopic/microscopic simulations in Case [2.67-2.67, 6] As an example in microscopic CFD simulations, we used three kinds of grid arrangements (fine, medium and coarse; see Table 1) to study the grid sensitivity for Case [2.67-2.67, 6]. Hang et al. [11] evaluated these CFD results using wind tunnel data in detail. Fig. 4a only shows an example vertical profile of velocity at Point V2 in Case [2.67-2.67, 6]. Hang et al. [11] confirmed that microscopic simulations using the medium and fine grids are better than the prediction using the coarse grid, and both are nearly the same good in predicting the velocity profile (i.e. mean flows) and the shape of turbulence profile except that they overestimated turbulence near the windward street entry (x/B ¼ 0) which is the well-known deficiency of standard k-3 turbulence model. In addition, for Case [2-2.67, 9], Hang and Li [10] reported that although RNG k-3 model predicted turbulence near the windward entry better but did worse in predicting mean flows than the standard k-3 model. Considering both the solution accuracy of mean flows and the computational time, we selected the medium grid arrangements and standard k-3 model as the default choices for all microscopic CFD simulations. By volume averaging microscopic CFD results in Case [2.67-2.67, 6], Fig. 4bed shows the grid independence of spatially-averaged (macroscopic) properties in horizontal thin slabs, and Fig. 4e,f displays sectional Forchheimer coefficients (CF (z)) in six REV units using fine grids (CF1(z)) and medium (CF2(z)) grids calculated by Eqs.

Fig. 6. In Case [2.67-2.67, 6]: Horizontal profiles of uf at different heights of (a) z ¼ 1.33B, (b) z ¼ 0.05B and (c) at the roof level of z ¼ 2.67B. (d) Horizontal profiles of wf at the roof level of z ¼ 2.67B.

50

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

Fig. 7. In Case [2.67-2.67, 6]: (a) Turbulent kinetic energy in y ¼ 0 in 3D microscopic simulations and (b) macroscopic turbulent kinetic energy (kf) in 2D porous regions as CF1(z) used. (cef) Effects of TKEdarcy, TKEForch, udarcy, uForch terms in 2D porous regions as CF0 ¼ 0.22 used.

4

Spatially averaged stream-wise velocity f u in 18 REV units In Case [2-2, 18] at z=0.05B at z=B at z=2B at z=0.05B In Case [2.67-2.67,18] at z=B at z=2.67B

0.6

Spatially averaged vertical velocity f w in 18 REV units at z=0.05B In Case [2-2, 18] at z=B at z=2B at z=0.05B In Case [2.67-2.67, 18] at z=B at z=2.67B

0.5 0.4 0.3

f

2

f

u (m /s)

3

b

w (m /s)

a

1

0.2 0.1 0.0

0

0

4

8

12

16

20

24

28

32

36

-0.1

0

4

8

x/B centre of REV units

c

20

24

28

32

36

x/B centre of REV units

Spatially averaged TKE k in 18 REV units In Case [2-2, 18] at z=0.05B at z=B at z=2B In Case [2.67-2.67,18] at z=0.05B at z=B at z=2.67B

0.8

2

2

16

f

1.2 1.0

k (m /s )

12

f

0.6 0.4 0.2 0.0

0

4

8

12 16 20 24 28 x/B centre of REV units

32

36

Fig. 8. Spatially-averaged (macroscopic) properties at different heights in Case [2.67-2.67, 18] and Case [2-2, 18] by volume averaging microscopic simulation results: (a) uf, (b) wf, (c) kf.

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

(14) and (15). Fig. 4bed confirms that, for microscopic CFD results with the medium and fine grids, although spatially-averaged stream-wise velocity (uf) and turbulent kinetic energy (kf) change little, however the relative change of spatially-averaged pressure (pf) is rather large. As a result, in Fig. 5e,f, CF1(z) using the fine grid are about 3.5 times of CF2(z) using the medium grid verifying that CF(z) is of grid sensitivity in this case. It is a shortcoming of present ‘Ventair’ code with RANS k-3 turbulence model and further investigations with large eddy simulations are required to provide highquality and reliable distribution of CF(z). We also find that CF(z) varies at different heights and in different REV units, showing that the constant Forchheimer coefficient (CF0 ¼ 0.22 as porosity f ¼ 0.75) by Eq. (13b) may be not sufficiently exact. The first building generates the largest form drag to the approaching wind and the second building produces the smallest form drag due to the shelter effect by the first building. The form drag increases a little from building No. 2 to No. 6. In addition, CF0 from Ergun equation (0.22) may provide an reference evaluation of CF(z) using different grids. CF2(z) using the medium grids is far less than 0.22 for most REV units except unit 1 and CF1(z) using the fine grids for units of No. 3e6 is more near to 0.22 (CF0). It can be predicted that CF1(z) using the fine grids should be better than CF2(z) using the medium

51

grids. To verify this, we used three kinds of Forchheimer coefficient (CF1 (z), CF2 (z) and CF0) in the following macroscopic simulations using porous turbulence model in Case [2.67-2.67, 6]. For microscopic CFD simulations of Case [2.67-2.67, 6], Fig. 5a shows 2D streamline and distribution of velocity in the planes of y ¼ B and z ¼ B, and Fig. 5b shows 2D streamline and distribution of stream-wise velocity and vertical velocity in y ¼ 0 (the symmetry plane of the main street). In microscopic simulations, three dimensional (3D) vortex structures exist in the secondary streets (see Fig. 5a) where wind is relatively weak. Along the main street, the stream-wise velocity decreases because there is an upward flow (i.e. positive vertical velocity, see Fig. 5b). In the macroscopic simulation using the porous turbulence model (see Fig. 5c), the above microscopic flow information between buildings is lost, and the flow is represented as a macroscopic velocity reduction and positive macroscopic vertical velocity (i.e. upward flow) through 2D porous regions. Fig. 6 displays horizontal profiles of macroscopic stream-wise velocity (uf) and vertical velocity (wf) at different heights using CF1(z) (from micro-CFD results using the fine grid), CF2(z) (from micro-CFD results using the medium grid) and CF0 (constant of 0.22) compared with spatially-averaged quantities from microscopic simulation results. Macroscopic simulations

Fig. 9. In Case [2-2.67, 9] with microscopic CFD simulation results with the medium grid: (a) 3D stream-line and vertical velocity in y ¼ B, (b) CF(z) in nine REV units using Method 2, (c) u and k in y/B ¼ 0. Macroscopic simulation results: (d) Distribution of uf and kf in 2D porous regions.

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

We studied Case [2-2.67, 9] with a building height variation using both microscopic and macroscopic simulations. In Fig. 9a, both 3D streamline and vertical velocity in y ¼ B (the symmetry plane of secondary streets) display that, a downward flow (i.e. negative vertical velocity) exists in front of the taller buildings and an upward flow (i.e. positive vertical velocity) appears behind the taller buildings. Fig. 9b shows the sectional Forchheimer coefficient CF(z)in nine REV units calculated by microscopic simulations with the medium grid using REV definition of Method 2. It is found that the form drags by the first building (unit 1) and the taller buildings

a

f

u in C ase [2-2.67,9](R EV m ethod2) at z=0.05B M icro C FD M acro C FD C F (z) at z=B M icro C FD M acro C FD C F (z)

3

f

Stream-wise velocity u (m /s)

4

2

1

0

-4

b

5

4

8 x/B

12

16

20

f

4 3 2 1 0

-4

c

0

u in C ase[2 -2 .6 7 ,9 ](R E V m eth o d 2 ) at z= 2 B M icro C F D M acro C F D C F (z) at z= 1 .3 3 B M icro C F D M acro C F D C F (z)

f

Stream-wise velocity u (m /s)

3.1.2. Effect of uniform building heights on macroscopic quantities In Fig. 8aec, using microscopic simulation results, we studied the effect of uniform building heights on horizontal profiles of spatially-averaged (macroscopic) flow quantities at different heights in Case [2.67-2.67, 18] and Case [2-2,18] using REV definition of Method 1(see Fig. 3a). It is easy to find that the macroscopic stream-wise velocity (uf) at the pedestrian level (z ¼ 0.05B, i.e. 1.5 m in full scale) and at z ¼ B in the lower array is smaller than that in the taller one, representing that the taller array may capture more air flushing its canopy layer. In the lower array, Fig. 8a also shows that uf at the roof level (z ¼ 2B) of the lower array increases from x/ B ¼ 29 to 35 where the negative macroscopic vertical velocity (wf) at the roof level (z ¼ 2B) appears in the same region (see Fig. 8b). wf in the taller array is always positive in the above three heights. Fig. 8c shows that turbulence at the roof level increases in the lower array (Case [2-2,18]) from x/B ¼ 20 to 35. Fig. 8b and c confirm that when uf in the downstream region of the lower array is sufficiently small, the external flow above the roof level tends to transport more air downwardly by the vertical turbulent shear stress and vertical mean flows which are represented by strong turbulence and a negative wf at the roof level. Finally Hang and Li [20] reported macroscopic stream-wise equilibrium state may appear in 14-row and 21-row cube arrays (H/W ¼ 1) in which a fully developed region exists where macroscopic quantities keep constant. But for the present 18-row highrise square building arrays (H/W ¼ 2 or 2.67), such macroscopic equilibrium state never appears since macroscopic flow variables always vary in the stream-wise direction.

3.2. Macroscopic/microscopic simulations in arrays with building height variations

1.4

0

4

8 x /B

12

16

20

f

w in C ase[2-2.67,9](R EV m ethod2) at z=2B M icro C FD M acro C FD C F (z) at z=B M icro C FD M acro C FD C F (z)

1.2 1.0

f

using CF1(z) may predict uf at z ¼ 1.33B and 0.05B (see Fig. 6a,b) and wf at z ¼ 2.67B (see Fig. 6d) generally well and it does a little worse in predicting uf at the roof level (z ¼ 2.67B,see Fig. 6c) than that at low levels. Finally, simulations using CF1(z) are the best in predicting macroscopic mean flows than CF2(z) and CF0, and those using CF0 are the second better and those using CF2(z) are the worst one, especially drag force modelled by CF2(z) seems far less than the actual form drag induced by buildings. Fig. 7a,b displays the distribution of turbulence in y ¼ 0 in 3D microscopic CFD simulations and in 2D macroscopic simulations using porous turbulence model. In microscopic simulations (Fig. 7a), turbulence near the windward edge (x/B ¼ 0) is large. In macroscopic simulations (Fig. 7b) turbulence near the windward edge of porous city region is relatively small and that near the top interface (z/B ¼ 2.67) is large. Since microscopic simulations using standard k-3 turbulence model is not good in predicting turbulence as a well-known deficiency, large eddy simulations (LES) are required to evaluate the macroscopic turbulence predictions using present porous turbulence model. The relative low macroscopic turbulence near the windward edge of porous city region can be explained by the effect of darcy terms and Forchheimer terms as CF0 (constant of 0.22) is used (see Fig. 7c,d). The terms of TKEdarcy pffiffiffiffi (2f2 mkf =K, in order of 0.01) and TKEForch (f3 ðrcF = K Þ ð8=3ÞQ f kf , in order of 1) are both negative, and TKEForch is large near the windward edge where the macroscopic velocity Qf is considerable which significantly reduces the macroscopic turbulence locally. Similarly in Fig. 7e,f, the term of uForch is about 100 times greater than the term of udarcy, verifying that Forchheimer terms always dominate the momentum reduction than the darcy terms in such high-Reynolds-number turbulent flow. These findings are similar with those in Hang and Li [20] for low-rise porous cube arrays. In a whole, distribution of CF1(z) seems better choice for macroscopic simulations in Case [2.67-2.67, 6] and present porous turbulence model seems a promising tool for predict macroscopic mean flows through high-rise building arrays.

Vertical velocity w (m /s)

52

0.8 0.6 0.4 0.2 0.0 -0.2

-4

0

4

8 x/B

12

16

20

Fig. 10. Horizontal profiles for microscopic and macroscopic simulation results at different heights in Case [2-2.67, 9]: (a) uf at z ¼ 0.05B and z ¼ B, (b) uf at z ¼ 1.33B and z ¼ 2B, (c) wf at z ¼ B and z ¼ 2B.

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

53

(unit 2, 4, 6, 8) are much stronger than those by the lower buildings (unit 3, 5, 7). Fig. 9c shows the microscopic stream-wise velocity and turbulent kinetic energy (k) in the symmetry plane of the main street (y/B ¼ 0). In macroscopic simulations using the porous turbulence model, Fig. 9d displays uf and kf in 2D porous media. It is found that uf in Fig. 9d decreases in a fluctuation shape due to the building height variation similar with u in Fig. 9c. Differences exist between Fig. 9c (micro-CFD) and Fig. 9d (macro-CFD) that, the turbulence near the windward boundary of the building array is large in Fig. 9c, but in Fig. 9d, the turbulence near the windward boundary of the porous region is small and those near the interfaces at the roof levels tend to be large. Fig. 10 shows the comparison of microscopic and macroscopic simulations results at different heights in Case [2-2.67, 9]. As displayed in Fig. 10a,b, in comparison to the microscopic simulation result, porous turbulence model predicted the macroscopic velocity profile at low levels (z ¼ 0.05B, B or 1.33B) relatively well, but it did worse near the interface (at z ¼ 2B). In addition, Fig. 10c shows that both simulations may predict that horizontal profiles of macroscopic vertical velocity are decreasing in a wave shape. In a whole, present porous turbulence model is promising in predicting macroscopic velocity profiles through an array with a height variation except those near the canopy interface, and it requires further investigations to evaluate the macroscopic prediction of turbulence.

stronger macroscopic urban airflows than lower ones with the same porosity if the total street length is effectively limited. Therefore, a better choice is to build relatively tall buildings and wide streets instead of relatively low buildings and narrow streets, i.e. to capture more airflow rates across windward boundaries and increase the porosity to obtain more wind pathways for rural wind to flow through. Meanwhile, to limit the total length of building arrays, it is suggested to use large open spaces such as gardens, squares, grasslands, natural waterways etc which can be named as wide ‘urban canyon’, to separate the long high-rise compact city (for example, city scale) into several shorter urban areas (for example, length of each is less than 1 km or neighbourhood scale). According to Oke [27], such ‘urban canyon’ had better to be sufficiently wide to make sure that the isolated roughness flow regime (IRF, W > 6H) appears in them.

4. Conclusions

References

Both microscopic CFD simulations using RANS k-3 turbulence model and macroscopic simulations using the porous turbulence model were carried out to study urban airflows in some idealized high-rise square building arrays with a parallel approaching wind. Building arrays with a uniform building height or a canopy height variation were considered. In comparison to microscopic simulation results which were validated by wind tunnel data, the porous turbulence model is promising to predict the macroscopic mean flows well except those near the interfaces (or canopy roofs) if suitable porous parameters are modelled, but it requires further investigation (for example, large eddy simulations) to evaluate the prediction of turbulence by the porous turbulence model, especially for the relatively low turbulence near the windward boundary of the porous array due to the sink effect of Forchheimer terms. Some flow information is neglected using porous turbulence model but the calculation load can be effectively reduced. Forchheimer terms always dominate the reduction of velocity and turbulence through high-rise porous city regions. Further investigations are still required before the present porous turbulence model can be applied for studying macroscopic airflows in real cities with various wind directions. Urban-type high-rise building arrays usually produce strong resistances to the approaching wind. So both kinds of simulations confirm a quick velocity reduction through high-rise porous building arrays. For arrays with uniform building heights (H ¼ 2B or H ¼ 2.67B, 4 ¼ 0.75), if the building array is not too long (for example, 18 rows, L ¼ 35B, 1.05 km at full scale), the taller building array (H ¼ 2.67B) may produce a larger macroscopic velocity profile than the lower one (H ¼ 2B). For flows through building arrays with a building height variation (H ¼ 2B or 2.67B), macroscopic velocity components decrease in a wave shape, and those with uniform building heights decrease smoothly. For some crowded urban areas like Hong Kong, many high-rise buildings have to be built surrounded by relatively narrow street networks to provide sufficient residential area in highly-populated regions. Wider streets (i.e. greater porosities) are thought better for city ventilation. Moreover, taller building arrays may experience

[1] Chan CK, Yao XH. Air pollution in mega cities in China. Atmos Environ 2008; 42(1):1e42. [2] Giridharan R, Lau SSY, Ganesan S, Givoni B. Urban design factors influencing heat island intensity in high-rise high-density environments of Hong Kong. Build Environ 2007;42(10):3669e84. [3] Britter RE, Hanna SR. Flow and dispersion in urban areas. Ann Rev Fluid Mech 2003;35:469e96. [4] Balczó M, Gromke C, Ruck B. Numerical modeling of flow and pollutant dispersion in street canyons with tree planting. Meteorol Z 2009;18(2): 197e206. [5] Gromke C, Buccolieri R, Di Sabatino S, Ruck B. Dispersion modeling study in a street canyon with tree planting by means of wind tunnel and numerical investigations e Evaluation of CFD data with experimental data. Atmos Environ 2008;42(37):8640e50. [6] Xie XM, Huang Z, Wang J. The impact of urban street layout on local atmospheric environment. Build Environ 2006;41(10):1352e63. [7] Chang CH, Meroney RN. Numerical and physical modeling of bluff body flow and dispersion in urban street canyons. J Wind Eng Ind Aerod 2001; 89(14e15):1325e34. [8] Bady M, Kato S, Ishida Y, Huang H, Takahashi T. Application of exceedance probability based on wind kinetic energy to evaluate the pedestrian level wind in dense urban areas. Build Environ 2011;46(9):1834e42. [9] Tominaga Y, Mochida A, Yoshie R, Kataoka H, Nozu T, Yoshikawa M, et al. AIJ guidelines for practical applications of CFD to pedestrian wind environment around buildings. J Wind Eng Ind Aerod 2008;96(10e11):1749e61. [10] Hang J, Li YG. Age of air and air exchange efficiency in high-rise urban areas and its link to pollutant dilution. Atmos Environ 2011;45(31):5572e85. [11] Hang J, Li YG, Sandberg M. Experimental and numerical studies of flows through and within high-rise building arrays and their link to ventilation strategy. J Wind Eng Ind Aerod, in press. [12] Meroney RN. Characteristics of wind and turbulence in and above model Forests. J Appl Meteorol 1968;4(5):780e8. [13] Finnigan JJ. Turbulence in plant canopies. Annu Rev Fluid Mech 2000;32: 519e71. [14] Antohe BV, Lage JL. A general two-equation macroscopic turbulence model for incompressible flow in porous media. Int J Heat Mass Tran 1997;40(13): 3013e24. [15] Nakayama A, Kuwahara F. A macroscopic turbulence model for flow in a porous medium. J Fluid Eng 1999;121(2):427e33. [16] Getachew D, Minkowycz WJ, Lage JL. A modified form of the k-3 model for turbulent flows of an incompressible fluid in porous media. Int J Heat Mass Tran 2000;43(16):2909e15. [17] Pedras MHJ, de Lemos MJS. Macroscopic turbulence modelling for incompressible flow through undeformable porous media. Int J Heat Mass Tran 2001;44(6):1081e93. [18] Lien FS, Yee E, Wilson JD. Numerical modelling of the turbulent flow developing within and over a 3-D building array, Part II: mathematical Formulation for a distributed drag force approach. Bound-Lay Meteorol 2005;114(2): 245e85.

Acknowledgements The work in this paper is supported by a University Development Fund from the University of Hong Kong on Initiative of Clean Energy for Environment. The support from Prof Mats Sandberg in KTH research school, University of Gavle in wind tunnel measurements is highly acknowledged.

54

J. Hang, Y. Li / Building and Environment 49 (2012) 41e54

[19] Chandesris M, Serre G, Sagaut P. A macroscopic turbulence model for flow in porous media suited for channel, pipe and rod bundle flows. Int J Heat Mass Tran 2006;49(15e16):2739e50. [20] Hang J, Li YG. Wind conditions in idealized building clusters e macroscopic simulations by a porous turbulence model. Bound-Lay Meteorol 2010;136(1): 129e59. [21] Lien FS, Yee E. Numerical modelling of the turbulent flow developing within and over a 3-D building array, Part III: a distributed drag force approach, its implementation and application. Bound-Lay Meteorol 2005;14(2):287e313. [22] Belcher SE, Jerram N, Hunt JCR. Adjustment of turbulent boundary layer to a canopy of roughness elements. J Fluid Mech 2003;488:369e98.

[23] Conceal O, Belcher SE. Mean winds through an inhomogeneous urban canopy. Bound-Lay Meteorol 2005;115(1):47e68. [24] MacDonald R. Modelling the mean velocity profile in the Urban Canopy Layer. Bound-Lay Meteorol 2000;97:25e45. [25] Di Sabatino S, Solazzo E, Paradisi P, Britter R. A Simple Model for Spatiallyaveraged wind profiles within and above an urban canopy. Bound-Lay Meteorol 2008;127:131e51. [26] Ergun S. Fluid flow through packed columns. Chem Eng Prog 1952;48(2): 89e94. [27] Oke TR. Street design and urban canopy layer climate. Energ Buildings 1988; 11(1e3):103e13.