Analysing urban ventilation in building arrays with the age spectrum and mean age of pollutants

Building and Environment 131 (2018) 288–305

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Analysing urban ventilation in building arrays with the age spectrum and mean age of pollutants

T

G.E. Lau∗, K. Ngan School of Energy and Environment, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Age Age spectrum (distribution) Urban ventilation LES

This study investigates the applicability and relevance of the Green's function and age spectrum to urban ventilation. Ventilation in an idealised urban environment is examined by characterising pollutant dispersion from the urban canopy layer. Air flow over regular uniform and non-uniform building arrays is computed using largeeddy simulation and pollutant removal is analysed via the age spectrum for localised sources. The age spectrum represents the probability distribution function of pollutant or tracer ages. Since the age spectrum is obtained directly from the evolution equation for the passive scalar, it is more effective for flows with high intermittency. This should be beneficial for studies of extreme pollution events in the urban environment such as accidental release of toxic gases and dust storms. It is demonstrated that the age spectrum is sensitive to the source release location, implying that pollutant removal depends on the initial conditions. Mean ages calculated using the homogeneous emission method and the age spectrum show qualitative similarities; however, there are quantitative differences in regions where the flow is highly unsteady and intermittent. The age spectrum indicates that ventilation decreases by ∼70% when the aspect ratio (building height/street width) is increased from 1 to 4. Furthermore, the effect of fresh air entrainment increases by ∼80% when the building heights are non-uniform.

1. Introduction Air pollution has become an important environmental issue in recent decades. In the urban environment, local air pollution typically originates from vehicular emissions [1]. Particulate matter (PM) and gaseous pollutants in built-up urban areas constitute a major long-term health risk to children [2], seniors [3] and people suffering from respiratory diseases [4–6]. Apart from local air pollutants, severe air pollution episodes such as accidental release of toxic gases, dust storms and bushfire smoke [7–9] also pose acute risk to the environment and health of urban inhabitants. Ventilation, which refers to the process by which pollutants are removed by fluid flow, is thus very important for both outdoor and indoor air quality [10]. It is known that urban ventilation and pollutant dispersion are affected by factors such as building packing density [11], street width [12, 13] and inflow wind conditions [14, 15]. Numerous field studies [16], experimental work [17, 18] and numerical simulations [19–25] have mainly focused on idealised domains, e.g. street canyon [19–23], 26, 27] and building arrays [11, 28–31] (see for example the reviews by Vardoulakis et al. [32] and Belcher [33]). It has been found that urban street canyons are often linked to poor ventilation with high pollutant concentrations. In particular, the presence of closely packed tall

∗

buildings causes the air flow to stagnate, greatly hindering the removal of pollutants from the urban canopy layer (UCL) [17, 34–36]. In studies of urban ventilation and pollutant dispersion, it is common to adopt flux-based indices such as the volumetric flow rate and air change rate [30, 37], pollutant dispersion and exchange rate [19, 22, 26], and exchange velocity and city breathability [16, 38, 39]. As pointed out by Bady et al. [34], these flux-based indices mostly correspond to the mean exchange across the UCL boundary, rather than the local ventilation within the UCL. An alternative perspective comes from building ventilation theory commonly applied to evaluate indoor ventilation [34, 40] where timescale-based indicators have been used to characterise ventilation within the urban environment. An example is the local mean age of air τair [29, 41, 42] which can be used to examine how rural air is supplied and distributed within an UCL. Large values implies a poorly ventilated region and vice versa. To estimate the local mean age of air, the homogeneous emission method [41] which assumes a spatially uniform source is commonly applied: the local mean age of air is estimated from the time-mean concentration, thereby establishing a link between the pollutant concentration level and its removal timescale. For a realistic urban area, the removal of pollutants is influenced by variability in the urban morphology [43], human activity [44] and

Corresponding author. E-mail address: [email protected] (G.E. Lau).

https://doi.org/10.1016/j.buildenv.2018.01.010 Received 13 October 2017; Received in revised form 4 January 2018; Accepted 4 January 2018 Available online 09 January 2018 0360-1323/ © 2018 Elsevier Ltd. All rights reserved.

Building and Environment 131 (2018) 288–305

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Nomenclature c g H H1 H2 k p* Qc ReH S1 S2 Sc Tc t (u, v, w )

uo u∞ Vc W x y z Z Zy Zo Δ Δτrel ξ κ

building area density kinematic viscosity modified perturbation pressure spanwise vorticity density strain tensor mean tracer age mean age (homogeneous emission) difference between inlet-source and ground-source mean age relative difference between τmta and τhom characteristic timescale of exponential decay canyon or channel averaged field spanwise averaged field

λp ν π∗ Ωy ρ Sij τmta τhom Δτ

concentration field gravity building height shorter building height (for non-uniform case) taller building height (for non-uniform case) turbulent kinetic energy perturbation pressure scalar source Reynolds number based on the building height street width (primary) street width (secondary) homogeneous source canyon circulation timescale time velocity in the streamwise, spanwise and vertical direction, respectively roof-level streamwise velocity free-stream velocity volume of a canyon or channel unit building width horizontal direction vertical direction span-wise or lateral direction age spectrum spanwise-averaged age spectrum global maximum of the age spectrum grid spacing relative difference between τmta and τhom time lag scalar diffusivity

Δτrel I c y

Subscripts o sgs

initial state SGS component

Superscripts

−

time-averaged field filtered field fluctuating field

∼ ′

Abbreviations LES PALM RANS

large-eddy simulation Parallelized Large-eddy Simulation Model Reynolds-averaged Navier-Stokes

attempts to examine sensitivity of the age spectrum to different initial conditions by considering source release locations at the ground level and inlet.

external winds [45]. This implies that the airflow within these urban configurations is highly turbulent and possibly intermittent [20]. In addition, pollutant sources may be inhomogeneous particularly in cases of extreme pollution where the disturbances to the urban environment are essentially transient in nature. In these cases, evolution of the pollutant field and hence its removal timescale depend on the unsteady flow and source release locations (i.e. initial conditions). More generally, there exists a statistical variability in the pollutant removal. Such effects should be taken into account when evaluating pollutant removal timescales for inhomogeneous urban domains, yet studies examining this aspect of the problem are uncommon. Recently, Lo and Ngan [46] applied the Green's function and age spectrum [47,48] to study ventilation for a single street canyon. The age spectrum represents a statistical or probability distribution function of tracer (or pollutant) ages. In this approach, the mean tracer age is calculated directly from the evolution of the scalar field as the first moment of the age spectrum; it encapsulates all the fluid dynamical processes that affect the distribution of a passive tracer. Since this approach does not require time-averaging or a spatially homogeneous source, the implication is that it can be used for localised sources and as an indicator to study inhomogeneity in urban ventilation. In this article, when there is no risk of ambiguity, we shall refer to the mean age of pollutant as the mean tracer age. In this work, we aim to examine ventilation within inhomogeneous urban domains using the age spectrum and mean age of pollutants. The urban domains investigated are idealised city-like geometries with uniform and non-uniform aspect ratios (building height/street width). Applicability of the age spectrum in characterising ventilation is examined by comparing localised and homogeneous sources. Specifically, the flow conditions under which the age spectrum differs from the homogeneous emission method are discovered. The present work also

2. Methodology 2.1. Numerical model The computational model is based on the non-hydrostatic and incompressible Navier-Stokes equations. To adequately resolve the turbulent structures, a large-eddy simulation (LES) approach was adopted in which the large-scale eddies are resolved explicitly and the smallscale eddies are filtered and modelled. This filtering operation, denoted by a tilde (∼ ) , is defined as

φ͠ (x i , t ) =

∫D F (xi , xi′) φ (xi′, t ) dxi′,

(1)

where D is the flow domain, F is the filter function, and x j = (x , y, z ) are axes of the Cartesian coordinate system. The equations for the conservation of mass, momentum and passive scalar, implicitly filtered over a grid volume on a Cartesian grid, read as:

∂u͠ j ∂x j

=0

(2)

∼∗ ∂u͠ i u͠ j ∂τij ∂2u͠ ∂u͠ i 1 ∂π =− + ν 2i − + ∂x j ∂x j ∂t ρ0 ∂x i ∂x j

(3)

∂u͠ j c͠ ∂sj ∂2c͠ ∂c͠ =κ 2 − +S + ∂x j ∂x j ∂t ∂x j

(4)

∼) are the velocity components with loIn Eqs. (2)–(4), u͠ i = (u͠ , v͠ , w ∼∗ = p͠ ∗ + 2 ρ e is the cation x i = (x , y, z ) , ρ0 is the density of dry air, π 3 0 289

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modified perturbation pressure with p͠ ∗ being the perturbation pressure 1∼ and e = 2 ui" ui" is the subgrid-scale turbulent kinetic energy where ui″ = ui − u͠ i is the subgrid-scale velocity. In Eqs. (3) and (4), τij = ∼ ui uj − u͠ i u͠ j and sj = ∼ uj c − u͠ j c͠ are the subgrid-scale (SGS) stress tensor and the SGS scalar flux vector, respectively. Because the small scales tend to be more isotropic than the large ones, their effects can be modelled by SGS-viscosity and diffusivity models which are written as

τij −

1 1 τkk δij = −2νsgs ⎛S͠ ij − S͠ kk δij ⎞ 3 3 ⎝ ⎠

t t ∂τijt ⎞ ∂2u͠ t ⎛ ∂u͠ i u͠ j +Δt u͠ itpred = u͠ it + ⎜− + ν 2i − ⎟ ∂x j ∂x j ⎠ ∂x j ⎝

Subsequently, the perturbation pressure is obtained from the Poisson equation as t+Δt ∼t ρ0 ∂u͠ ipred ∂2p∗ = Δt ∂x i ∂x i2

∼t ⎛ 1 ∂p∗ ⎞ +Δt u͠ it + Δ t = u͠ itpred + Δt ⎜− ⎟ ⎝ ρ0 ∂x i ⎠

∂s͠ ∂x j

1 2

(

∂u͠ i ∂xj

+

∂u͠ j ∂xi

) is the strain tensor.

2.2. Age spectrum and mean tracer age

In the present paper, the Deardorff [49] SGS model as modified by Moeng and Wyngaard [50] and Saiki et al. [51] is used, wherein the SGS viscosity and scalar diffusivity are related to the SGS turbulent kinetic energy, viz.

νsgs = cm l e

The application of tracer age in the study of urban canopy ventilation is illustrated in Fig. 1. Consider an air parcel that is marked with a tracer and given a ‘clock’ that is started at the source location, → x 0 at a time to. Here, an air parcel is imagined to have finite but arbitrarily small volume in the continuum limit. Now, assume the tracer parcel reaches an arbitrary receptor location → x at time t = t0 + ξ where ξ is the time lag or tracer age. As the pathway connecting the source and receptor depends on the turbulent flow, there exists a statistical distribution of ages, or age spectrum, connecting the source and receptor locations. The age spectrum thus provides information regarding the statistical variability of pollutant ventilation in the urban environment. Tracer age spectrum was mathematically derived by Holzer and Hall [48]. To apply it in the context of the urban environment, the tracers are modelled as passive scalars [46]:

(7)

and

κsgs = (1 + 2l/Δ) νsgs

(8)

Here, Δ = (Δx1 Δx2 Δx3)1/3 is the volume-averaged grid spacing and l = min(1.8z , Δ) is the SGS mixing length. The Smagorinsky coefficient cm = 0.1 as is common for studies of shear flows [52–57]. The closure of Eqs. (7) and (8) requires a governing equation for the SGS turbulent kinetic energy:

∼ ∼ ∂u p″ ⎞⎤ ∂e ∂e ∂ ⎡ ⎛ + uj = −(ui″ u″j ) i − ⎟⎥ − ε ⎢u″j ⎜e + ρ0 ⎠⎥ ∂t ∂x j ∂x j ∂x j ⎢ ⎝ ⎦ ⎣

(12)

Eq. (12) is solved using an iterative multigrid method. The reader is referred to Maronga et al. [58] for more details on the implementation of the numerical schemes.

(6)

In Eqs. (5) and (6), νsgs is the SGS viscosity, κsgs is the SGS scalar diffusivity and S͠ ij =

(11)

In the corrector step, the velocity field is corrected using Eqs (10) and (11):

(5)

and

sj = −κsgs

(10)

→ ∂c u · ∇ c = κ∇2 c + S +→ ∂t

(13) → where c is the passive scalar concentration, u is the velocity vector, κ is the scalar diffusivity, t is time and S is the scalar source. As Eq. (13) is x , t→ x 0 , t0) such that the solution linear, it admits a Green's function G (→ takes the form

(9)

where ε is the SGS dissipation. Near the buildings, a wall model is required to accurately simulate the dynamics of the shear flow. In the present work, the wall model assumes a constant flux Prandtl layer at each building surface (vertical and horizontal). The governing equations Eqs. (2)–(4) and Eq. (9) are solved using the Parallelized Large-eddy Simulation Model, PALM code [58]. The advection terms are discretised using an upwind-biased fifth-order differencing scheme while the diffusion terms are discretised using second order central difference. Time-stepping is achieved via a thirdorder Runge-Kutta scheme [59]. Pressure-velocity coupling is implemented using a predictor-corrector method where an equation is solved for the modified perturbation pressure after every time step. In the predictor step, Eq. (3) is solved without the pressure term, giving a + Δt predicted velocity field u͠ itpred at time level t + Δt as

t

c (x , t ) =

∫ dt0 ∫ G (→x , t →x 0 , t0) S (→x 0 , t0) dx→ 0

Ω

(14)

where Ω denotes the region over which the scalar source S is applied. Essentially, the Green's function is a solution of Eq. (13) with a deltafunction source, viz.

→ ∂G +→ u · ∇ G = κ∇2 G + δ (→ x −→ x 0) δ (t − t0) ∂t

(15)

From Eqs. (14) and (15), it can be shown that a probability Fig. 1. Illustration of the concept of age of pollutants in an inhomogeneous urban canopy layer.

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distribution of tracer ages is given by

Z (→ x , ξ) =

Table 1 Summary of the mesh parameters used in this study.

→ x , t→ x 0 , t − ξ ) S (→ x 0 , t − ξ ) dx ∫ G (→ 0 Ω t

→ x , t→ x 0 , t0) S (→ x 0 , t0) dx ∫ dt∫ G (→ 0 0

Ω

Case U1 Case U2, N1 Case U3, N2

(16)

Eq. (16) is normalized such that ∫ Zdξ = 1. In this work, the Green's function is calculated for a single initial release time and the scalar source S is an area source. The age spectrum Z represents the fraction of tracer mass residing in volume Ω that has been in the flow for time t − t0 . There is relatively good pollutant removal capacity for a distribution skewed towards small values of ξ and relatively poor removal for one skewed towards large values of ξ . Also, advection dominates when the distribution has a sharp and narrow peak, while diffusion plays a more important role when there is a broad distribution. Following from this, the mean tracer age τmta can be defined by the first moment of Z , viz.

Δx × Δy × Δzmin

Total number of grid points

1m×1m×1m 1m×1m×1m 1m×1m×1m

1.8 × 107 2.6 × 107 3.2 × 107

Table 2 Summary of the geometrical parameters examined in this study.

Case Case Case Case Case

U1 U2 U3 N1 N2

H/W

W/S1

W/S2

H1/W

H2/W

1.00 2.67 4.00 N/A N/A

1.0 1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0

N/A N/A N/A 2.00 2.67

N/A N/A N/A 2.67 4.00

∞

τmta =

∫ ξZdξ 0

by calculations of the age spectrum and mean tracer age are comparable to those using conventional ventilation indicators.

(17)

In principle, τmta represents the average time required for the tracer to travel from the source at → x . Small values x 0 to an arbitrary location → of τmta generally imply good pollutant removal capacity whereas large values of τmta may be associated with poor ventilation and longer transit time. Here, ventilation refers to the cumulative removal of pollutants from the volume Ω. It differs from ‘ventilation efficiency’ which is generally related to the decay rates [60]. The simulation times required

3. Model configurations The computational domains are illustrated in Fig. 2. The parameters are summarised in Tables 1 and 2. Uniform (Fig. 2(a)) and non-uniform (Fig. 2(b)) building arrays are investigated. As may be seen from Fig. 2, the x, y, and z-directions represent the streamwise, spanwise and Fig. 2. Computational domains: (a) uniform building arrays with height H, and (b) non-uniform building arrays with height H1 (empty) and H2 (shaded). Buffer region at the outflow is not shown here.

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Eq. (14) from this data. For all simulations, stability of the numerical algorithm was ensured by employing a maximum Courant number of 0.35 in all simulations with a maximum time step of 1 × 10−2 s . The divergence of the three-dimensional velocity field, which is reduced iteratively by the pressure solver, has a maximum residual of 1 × 10−6 .

vertical directions, respectively. The approaching wind is parallel to the primary streets (street width S1 in Fig. 2) and perpendicular to the secondary streets (street width S2 in Fig. 2). Also, H = 20 m denotes the building height, W denotes the building width and the building area density λp = W × W /[(W + S1)(W + S2)] = 0.25. The computational domain extends to 24W, 10W and 8H in the x, y and z-directions, respectively. The vertical height of 8H has been shown to adequately capture the roughness sublayer [28]. A buffer region extending to 5W is included for numerical stability. In the case of uniform building arrays, three H/W aspect ratios are considered in this paper, i.e. H/W = 1 (Case U1), H/W = 2.67 (Case U2) and H/W = 4 (Case U3). Meanwhile, two non-uniform cases are considered in this paper: (i) H2/H1 = 1.34 (Case N1) and H2/H1 = 1.50 (Case N2). In order to label their positions within the urban canopy, the building arrays are divided into finite repeating units following the notation used by Hang and Li [29]. As illustrated in Fig. 2, the main streets are composed of ‘channel units’ while the secondary streets are composed of ‘canyon units’. Canyon and channel units are designated by their indices, i.e. 1 (upstream), 2, 3, …, and 8 (downstream). As shown in Table 1, the grid spacing within the urban canopy is 1 m × 1 m × 1 m. Above 4H, the wall-normal grid spacing is expanded geometrically by a factor of 1.015 across adjacent grid points. It has been shown that this resolution is sufficient to resolve the turbulent features within the urban canopy layer [29,41,61,62]. As in Letzel et al. [20], an isothermal atmosphere was forced by a streamwise pressure gradient of −0.0006 Pa/m. For case U1, the characteristic streamwise velocity u 0 = 1.6 ms−1 at the roof level and the Reynolds number ReH = uo H / ν is greater than 4000, implying that the flow is fully turbulent [63,64]. Free-slip boundary conditions for the velocities are applied at the top and spanwise boundaries. No-slip is used at the bottom boundary. Dirichlet and radiation boundary conditions are used at the inflow and outflow boundaries, respectively. Turbulence at the inlet is maintained by applying the turbulence recycling method as described by Lund et al. [65] and modified by Kataoka and Mizuno [66]. The source flux is specified at either the ground level or inlet. The model was spun up for approximately 21600 s. Subsequently, data are collected for 7200 s; the Green's function is calculated based on

4. Validation To validate the flow field in the present numerical simulations, we refer to two wind tunnel experiments that were carried out for different idealised urban geometries. Uniform building arrays of aspect ratio H/ W = 1 and building area density λp = 0.25 were investigated by Uehara et al. [64]. In their wind tunnel experiment, the approaching wind was parallel to the main streets and the Reynolds number ReH = uo H / ν was at least 4000. The flow field within and above the street canyon was measured using a laser Doppler anemometer (LDA). Wind tunnel experiments for uniform and non-uniform building arrays were also performed by Hang [67]. In this experiment, regular arrays of cubes were placed in a 11-m long wind tunnel and the approaching wind was also parallel to the main streets as in the experimental setup of Uehara et al. [64]. The building area density λp was also 0.25, but the height aspect ratio H/W was greater, i.e. 2 and 2.67. Normalised velocity profiles and turbulence quantities are compared with the respective wind tunnel data. Case U1 is compared with the wind tunnel data of Uehara et al. [64] (Fig. 3) while cases U2 (Fig. 4) is validated against the data of Hang [67]. In these figures, the mean velocity is normalised by the bulk wind u∞ and 1 k = 2 (u′2 + v′2 + w′2) is the turbulent kinetic energy where u′i = ui − ui is the velocity fluctuation. For case U1 (Fig. 3), very good agreement is observed between the current LES results and the experimental data. The LES is able to accurately predict the profiles of the mean u-velocity and turbulent kinetic energy. Agreement is worse for the velocity fluctuations. Slight discrepancies are also observed in the shear layer where the bulk flow separates into the canopy. Comparing the present LES results for case U2 (Fig. 4) with the wind tunnel data of Hang [67], we find that the agreement is very good for the mean velocity. Agreement for the mean turbulent kinetic energy is fairly good within the canyon unit but worse above the canyon units. It Fig. 3. Vertical profiles at x/W = 15 and y/W = 0 for case U1: (a) normalized u-velocity, and (b) normalized rms u-velocity fluctuations.

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Fig. 4. Vertical profiles at x/W = 15 and y/W = 0 for case U2: (a) normalized u-velocity, and (b) normalized turbulent kinetic energy k.

source. In addition, a closer examination of Fig. 6(a), (b) and (c) indicates that there is a poorly-ventilated region (as shown by the large mean age) close to the roof level where the unsteady shear layer separates. Thus, this region which is generally characterised by the recirculation zone has a tendency to increase pollutant concentration [71]. Increasing the building height results in larger vertical changes in the magnitude of the mean tracer age (Fig. 6(b) and (c)). The vertical profiles for the uniform building arrays (i.e. cases U1, U2 and U3) differ qualitatively from those obtained by Lo and Ngan [46] for the case of skimming flow in a single street canyon. In their study, the mean flow is characterised by a canyon vortex where the mean age is large. In this work, however, the mean tracer age is seen to be much more uniform most likely because the flow in the canyon unit

is useful to compare the results with the RANS results of Hang and Li [29] which are shown in the respective figures. The RANS results show greater discrepancies when compared to the wind tunnel data, especially in the shear layer where the flow is highly inhomogeneous. This suggests that the flow is highly unsteady. As we shall show in the results that follow, this could explain the differences between the homogeneous emission method and the present approach. The scalar fields are also validated against wind tunnel measurements. For consistency with the experiments of Meroney et al. [68], Pavageau [69] and Pavageau and Schatzmann [70], a line source is used in place of the area source for the simulations. In the wind tunnel experiments, a street canyon of aspect ratio H/W = 1 was investigated. The approaching wind was perpendicular to the street canyon while the continuous ground-level line sources were parallel to the canyon axis. Fig. 5 compares the dimensionless mean pollutant concentration cu∞ H / q along the leeward and windward walls of canyon 5 for case U1 with the wind tunnel measurements. Here, q is the emission rate per unit length along the centerline of the canyon. As shown in the figure, the agreement between the LES results and the wind tunnel data is reasonably good at the roof level and in the core of the canyon unit. Slight discrepancies are found in regions close to the ground. 5. Results and discussion 5.1. Pollutant dispersion for a ground-level source In this section, we explore the characteristics of pollutant removal from the urban canopy layer by examining the mean tracer age and the age spectrum. The passive scalar source flux is specified at ground level and spans the computational domain. 5.1.1. Mean tracer age Fig. 6 plots the mean tracer age τmta in the central x-z plane, i.e. at y/ W = 0.0, for all cases studied in this paper. Generally, τmta increases almost uniformly with height in each canyon unit; it is small near the ground level and increases towards the roof level. This reflects the fact that more time is required for tracer parcels to travel farther from the

Fig. 5. Vertical profiles of the dimensionless mean concentration cu∞ H / q at x/W = 15 and y/W = 0 for case U1 on the windward (- -) and leeward (–) walls. The wind tunnel data were taken from Pavageau [69] (circles), Meroney et al. [68] (squares) and Pavageau and Schatzmann [70] (triangles). Filled symbols represent the values on the leeward wall whereas empty ones represent the values on the windward wall.

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that the mean age of pollutants is not homogeneous in the streamwise direction; generally it increases downstream. For case U3, the increase between canyon 1 and canyon 8 is ∼62.5%. One possible explanation is that pollutants released inside upstream canyon units could re-entrain into the downstream canyon units, thereby increasing the mean age. As the source flux is not localised inside specific canyon or channel units, the contribution of the re-entrained pollutants to the mean tracer age is difficult to quantify. Contribution of the re-entrained pollutants may be examined by other Lagrangian-based indicators such as the residence time [72,73]. This is not examined in this paper; further studies examining this aspect of the problem will be conducted in the future. Meanwhile, comparing the canyon units of the uniform building arrays, we find that the maximum difference in the mean pollutant ages between cases U1 (canyon 8) and U3 (canyon 8) is ∼70%. As will be shown in section 5.4 of this article, removal of pollutants becomes much more difficult downstream, especially for deep urban canopies such as the one in case U3, because the flow is generally weaker. This phenomenon, when coupled with the re-entrainment of upstream pollutants, serves to increase the mean tracer age and degrade ventilation. As expected, non-uniformity in the building heights enhances ventilation and the mean dispersion of pollutants [11,29]. It is clear from Figs. 6 and 7 that the mean pollutant ages for cases N1 and N2 are much lower than their uniform counterparts i.e. cases U2 and U3. On average, the mean pollutant age is lowered by 15% and 25% for cases N1 (H2/ H1 = 1.34) and N2 (H2/H1 = 1.50), respectively. This effect is more pronounced when the ratio H2/H1 is greater, for instance in case N2. Further discussions in section 5.4 will show that non-uniformity in the building morphology generates inhomogeneity within the canopy flow such that older pollutants are easily removed from the deep canyons. By contrast with the canyon units, ventilation generally improves in the channel units as they are parallel to the approaching wind. However, as evident in Fig. 7, the improvement is relatively small. As the local flow is stronger in the channel units, this reinforces the idea that upstream pollutants re-entrain into the downstream units, thereby increasing the mean pollutant age. 5.1.2. Age spectrum Fig. 8 shows the age spectrum Z y near the centre of the domain (i.e. within canyon 5) for case U1; the heights chosen are z/W = 0.1 (pedestrian level), 0.5 (canyon mid-height) and 1.0 (roof level). Here, y denotes spanwise-averaging within a canyon or channel unit, viz.

Z Fig. 6. Mean tracer age τmta in the central x-z plane: (a) case U1, (b) case U2, (c) case U3, (d) case N1, and (e) case N2. Pollutant source release locations are represented by red lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

c

=

1 Vc

∫V Z dxdydz c

=

1 Ly

Ly

∫ Z dy 0

(19)

where L y is the spanwise length of a canyon or channel unit. At all heights, the age distributions are skewed towards lower values of ξ; the distributions exhibit global maxima for ξ ≤ 500 s after which they decrease exponentially towards zero. Exponential decay is a property of pure diffusion in an unbounded domain, but the behaviour for turbulent flow in an inhomogeneous domain with solid boundaries could differ. The peak around the global maximum is the sharpest at z/ W = 0.1 and broadens towards the roof level at z/W = 1. In addition, the location of this initial peak (e.g. the peak at ξ = 50 s for z/W = 0.1) changes with height. As pointed out by Lo and Ngan [46] for the case of a single street canyon, the location of the initial peak can be explained in terms of the transit time between the source and receptor; the pollutants require a longer transit time to reach locations that are farther from the ground. Following from this, the initial peak typically refers to the advective timescale while the exponential decay is likely related to spanwise averaging of the turbulent diffusion, which gives rise to diffusive-like behaviour [74]. The location of the initial peak and the shape of the profile are greatly affected by the urban geometry. A closer examination of Fig. 8 also reveals that there are secondary peaks (e.g. at ξ = 1500 for z / W = 1.0 ) along the exponential tail. These peaks indicate that ventilation of the canyon unit occurs on various timescales,

is much more homogeneous and the vortex is not prominent. When the building heights are non-uniform (Fig. 6(d) and (e)), we find that there is increased variability in the mean tracer ages and they are smaller compared to the uniform cases. The non-uniformity in building morphology improves ventilation by creating inhomogeneity in the flow. This is consistent with the findings of Lo and Ngan [46] for the case of non-uniform street canyons. To quantify the mean tracer age in different canyons or channels, we compute the canyon- or channel-averaged mean tracer age, τmta c as given by

τmta

y

(18)

where Vc is the volume of a canyon or channel unit c. Fig. 7 presents the canyon- or channel-averaged mean age of pollutants for all cases examined in this paper. Comparison with the results of Hang and Li [29] (not shown) indicates quantitative similarities. In each case, we find 294

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Fig. 7. Volume-averaged mean tracer age τmta c for a pollutant released at the ground level: (a) canyon units, and (b) channel units.

In this equation, Z0 is the global maximum and I is estimated from the inverse slope of the best-fit line for Z ≥ Z0 . The characteristic time I is a convenient ventilation indicator because it quantifies the rate at which aged pollutants are removed from the canopy. In general, 1/I differs from the rate at which the pollutant field decays. Furthermore, it does not depend on the initial peak which is generally related to the transit time. Fig. 10 summarises I for the different cases. We see that not only does I increases farther downstream, it is also much greater inside deeper urban street canyons. The maximum difference between the baseline case U1 and cases U2 and U3 is 37.5% and 50%, respectively (Fig. 10(a)). On the other hand, I is generally lower for the channel units; hence pollutant exposure is decreased (Fig. 10(b)). Removal of the passive pollutants from the channel units is more dominated by advection as the channels are parallel to the approaching wind. Turning now to the non-uniform cases N1 and N2, I has an interesting zigzag pattern along the streamwise direction. As observed previously [11,29], varying the building heights degrades ventilation in the canyons upstream of the taller buildings but improves ventilation in the downstream ones. The opposite pattern is observed for the channel units (Fig. 10(b)). This effect is now quantified. The ventilation rate shows a maximum improvement of 28% for case N1 and 45% for case N2 over to their uniform counterparts, i.e. cases U2 and U3.

Fig. 8. Age spectrum Z y at heights z/W = 0.1, 0.5 and 1.0 within canyon 5 of case U1. Pollutant is released at the ground level.

suggesting that there are different pathways by which pollutants reach the receptor locations. Fig. 9 compares the age spectra Z y near the centre of the domain (i.e. canyon 5 and channel 5) for all cases listed in Table 2. In this figure, Z y is taken at the roof level. The qualitative behaviour is consistent with previous studies [75]. For the canyon units (Fig. 9(a)), the age spectrum broadens as the building height is increased, indicating that ventilation deteriorates; for case U3, ventilation is the poorest because the advection within the urban canopy is the weakest. As non-uniformity in the building heights is increased, the slopes of the exponential tails steepen, indicating that ventilation is improved. For the channel units (Fig. 9(b)), the behaviour is similar. However, the secondary peaks along the exponential tail are stronger than the canyon units. This can be related to the dispersion of pollutants from canyon to channel units. For an approaching wind parallel to the primary streets, the flow within canyon units is more three-dimensional than that within the channel units, thereby leading to enhanced spanwise dispersion of pollutants from the canyon units. This is consistent with the results of Hang et al. [11]. The ventilation rate at long times can be quantified with 1/I , where I is the characteristic timescale of exponential decay. It is calculated from the inverse slope of the age spectrum [73,76]:

Z = Z0 exp(−ξ / I)

5.2. Effect of fresh wind entrainment on ventilation In this section, we consider a different aspect of urban ventilation. As pollutants can also be advected into the domain, we analyse a source at the inlet rather than the pedestrian level. This setup allows us to study the effect of fresh wind on pollutant dispersion within the urban canopy. 5.2.1. Age spectrum Fig. 11 shows roof-level age spectra for an impulse source located at the inlet. Broadly speaking, profiles of the age spectrum are qualitatively similar to those for the pedestrian-level source (Fig. 9): each has a global maximum after which the probability density function decreases exponentially. However, the onset of the exponential tail shifts as one progresses downstream from the inlet. This probably reflects differences in the transit time between the source (inlet) and the receptor (roof level). For a single street canyon with pollutant source at the ground level, Lo and Ngan [46] argued that the location of the global

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Fig. 9. Roof-level age spectrum Z y : (a) canyon 5, and (b) channel 5. Pollutant is released at the ground level.

(Fig. 6), τmta is small near x/W = 1 but considerably larger downstream e.g. at x/W = 15. Furthermore, there is marked asymmetry between the windward and leeward walls as the downward flow induces fresh air into the canyon units. A striking contrast with Fig. 6 is that the mean pollutant age does not necessarily increase with distance from the ground; in fact it is large in regions close to the ground. This is because more time is needed for pollutants released at the source to reach the ground level, in agreement with the time delay observed in the corresponding spectra (Fig. 11). The preceding implies that the age spectrum and mean tracer age are sensitive to the initial conditions of the localised source. The age spectrum, which is a statistical distribution of the tracer ages, depends on the source location; the further the receptor is from the source, the larger the mean tracer age is. For a bounded domain with homogeneous pollutant sources, for example in indoor ventilation problems,

maximum is related to the time required for pollutants to travel from the ground to the roof level. Similar behaviour is observed in Fig. 9 as the distance between the ground-level source and roof-level receptor is the same for all canyon units. In Fig. 11, by contrast, longer transit times are required for the pollutants to reach the downstream canyons. The foregoing is supported by an analysis of the transit times. The average transit time in all cases is ∼ 2nW / uroof where n denotes the canyon number and uroof is the time-averaged u-velocity at the roof level. For example, 2nW / uroof = 62.5 s for canyon 5 of case U1, which is consistent with the results shown in Fig. 11(a). This suggests that the initial delay is determined mostly by the mean advection.

5.2.2. Mean tracer age Fig. 12 plots the mean tracer age τmta in the central x-z plane (y/ W = 0.0) for the inlet source. As with the pedestrian-level source

Fig. 10. Characteristic time I of the age spectrum at roof level: (a) canyon units, and (b) channel units. I is not computed for channel 1 as the age spectrum does not have an exponential tail in this case.

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Fig. 11. Roof-level age spectrum Z y for each canyon unit: (a) case U1, (b) case U2, (c) case U3, (d) case N1, and (e) case N2. Pollutant is released at the inlet.

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Fig. 12. Mean tracer age τmta in the central x-z plane for the source at the inlet: (a) case U1, (b) case U2, (c) case U3, (d) case N1, and (e) case N2. Pollutant source release location is represented by red lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

mean tracer age of a pollutant released at ground level. Since τmta inflow and τmta ground relate to the transit time of pollutants from the source to the receptor, the difference between them should quantify the influence of the source location. When τmta inflow ≫ τmta ground , the dilution of pollutants from the urban canopy layer is driven by the flow inside the canopy. On the other hand, when τmta inflow ≪ τmta ground , the dilution of pollutants is driven by the advection of fresh external wind into canopy. The latter condition is much more desirable as it enhances pollutant transport and ventilation [75].

ventilation can be considered independent of the initial conditions [77]. Nevertheless, this may not be true for a domain with open boundaries and localised pollutant sources. In such a situation, which is typical for urban pollution problems, the removal of pollutant from the urban canopy will depend on initial conditions. For pollutants released at the inlet, the age spectrum and mean pollutant ages quantify the entrainment of fresh wind on pollutant transport within the urban canopy. Suppose that τmta inflow is the mean tracer age of a pollutant source specified at the inlet and τmta ground is the 298

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reaches ∼50% near the top of the canopy. Once again, the large difference for 17 ≤ x / W ≤ 18 is a boundary artefact. The structure of Δτrel can be attributed to several factors. First, τhom relies on a homogeneous source whereas τmta is based on a localised one. Second, τhom is calculated from the time-averaged concentration while τmta corresponds to a time-dependent quantity. Consequently, unsteady features of the flow and scalar fields may not be well-captured τhom . The larger Δτrel at the top of the canopy and along the downwind wall in Fig. 15 likely arise from highly unsteady flow in the shear layer. As will be demonstrated in the next section of this article, propagation of the instabilities in the form of Kelvin-Helmholtz waves renders the flow intermittent [20]. The upshot is that agreement between the two ages depends on the flow regime and the physical problem under examination. While τhom is robust and can be easily applied for flows that are relatively steady, τmta is more effective for flows that are highly unsteady or intermittent such as in cases of extreme pollution.

To illustrate the preceding, Fig. 13 presents the canyon- or channelaveraged difference Δτ = τmta ground c − τmta inflow c for all cases examined in this paper. Positive values indicate ventilation that is driven by the external wind whereas negative values indicate ventilation that is driven by the local flow within the canyon or channel units. As clearly indicated, the effect of the local flow is much more dominant in cases U1, U2 and U3 with case U3 demonstrating the strongest effect due to its high aspect ratio. For these cases where there is skimming flow and the building heights are uniform, the effect of fresh air on ventilation is minimal. The flow is generally weak (Fig. 7) and turbulent diffusion dominates advection (Fig. 10). Nonetheless, external winds have more effect on the channel units as they are parallel to the approaching wind. The external flow is more important when the building array heights are non-uniform. Since the magnitude of the vertical velocity increases with non-uniformity, the advection of fresh air into the canyon units is stronger in cases N1 and N2 than U2 and U3. This is also consistent with the age spectra (Fig. 10) where I is lower for N1 and N2. Increasing H2/H1 enhances the influence of the external winds on the canyon and channel units.

5.4. Relationship between mean tracer age and flow characteristics The flow within idealised street canyons and building arrays has been examined in great detail for low and moderate aspect ratios [11,19,22,26,28]. Here, we focus instead on the relationship between the flow characteristics and the pollutant removal timescales. For a single street canyon of unit aspect ratio, it is well known that there is skimming flow with a strong canyon vortex. One consequence of this is that the mean exposure and residence times are mostly determined by the mean flow or canyon circulation timescale [46]. However, it is not clear that this is true for an idealised building array configuration where the flow is inherently three-dimensional and inhomogeneous. Fig. 16 presents mean streamlines and mean turbulent kinetic energy in the central x-z plane (y/W = 0) for 10.5 < x/W < 16.5. As the flow above the urban canopy departs from the trailing edge of a building, it impinges upon the windward wall and creates a region of large k. This region is well known for its large velocity gradient and shear [30]. Since small-scale turbulence travels towards the ground in the form of Kelvin-Helmholtz waves [78,79], the mean age near the windward wall is lowered (Figs. 6 and 13). As the aspect ratio increases in cases U2 and U3, advection weakens towards the ground (see Fig. 16(b) and (c)). Small-scale turbulence is dissipated as it propagates down the windward wall. This is particularly evident in case U3 for which the mean streamlines are nearly parallel and k is very small for z/W < 2.5. For these cases, the mean flow may be separated into two distinct regimes: a recirculating flow close to the

5.3. Comparison with the homogeneous emission method The homogeneous emission method is a popular way of calculating the mean age of air (see Section 1). We now present an objective comparison of the mean tracer age τmta based on a source at the inlet (see section 5.2) with the mean age of air computed using the homogeneous emission method [41] τhom . In the latter, homogeneous pollutant sources were defined in the entire UCL as Sc = 1 × 10−5 kgm−1s−3 and

τhom =

c Sc

(21)

where c is the time-averaged concentration. Fig. 14 shows the profiles of τhom in the central x-z plane for cases U1, U2 and U3. At first glance, the τhom fields appear to be qualitatively similar to τmta (see Fig. 12); the mean age of air is large at ground level and also near the trailing edges of the buildings. Near the outflow boundary 17 ≤ x / W ≤ 18, τhom is smaller compared to the preceding canyon units. This is likely a boundary artefact: the flow in this region does not lie inside a canyon unit but rather within the wake of the last building. Fig. 15 plots the relative difference between τmta and τhom which is τ −τ indicated by Δτrel = mtaτ hom in the central x-z plane. Δτrel is not unihom form throughout the canyon units; it is small near ground level, but

Fig. 13. Volume-averaged difference Δτ between the mean tracer ages for inlet and ground-level sources: (a) canyon units, and (b) channel units.

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Fig. 14. Mean age of air, τ hom , calculated using the homogeneous emission method in the central x-z plane: (a) case U1, (b) case U2, and (c) case U3. Homogeneous pollutant sources are defined within the UCL as represented by the red lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

shear layer and a dissipative flow where the streamlines are parallel. To investigate the relationship between the mean tracer age and the mean flow, we normalise the former by the canyon circulation timescale for cases U1, U2 and U3. Following Lo and Ngan [73], the average H W canyon circulation timescale is Tc ≈ 2 w + u where u and w are the mean streamwise and vertical velocities inside the canyon unit. Table 3 shows that the mean pollutant age within uniform canyon arrays is not well-described by Tc ; for example, τage = 0.44Tc (case U1). The

(

agreement between the mean ventilation timescale and Tc is better for single street canyons [73]; τresidence = 1.26Tc for an idealised canyon with periodic boundary conditions in the spanwise direction, and τresidence = 0.78Tc for one defined by realistic urban geometry. Here, τresidence is the mean residence time (which differs from the mean age by excluding re-entrainment). Comparing the preceding cases, we find that the mean pollutant age in our calculations is significantly shorter, probably due to stronger turbulence.

)

Fig. 15. Relative difference Δτrel between τmta and τ hom at the central x-z plane: (a) case U1, (b) case U2, and (c) case U3.

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(

(H + H ) / 2

W

)

1 2 Tm ∼ + u . Table 4 shows that Tm gives a better estimate of w τmta than Tc for the non-uniform arrays. Nevertheless, the discrepancies suggest that there may be other factors influencing the mean pollutant age. The roof-level vorticity strongly influences the transfer of pollutants between the street canyon and the external flow. This has been demonstrated for a two-dimensional street canyon of unit aspect ratio [20,78]. The relationship between the mean age and roof-level vorticity is now investigated. Fig. 17 shows the time-averaged y-vorticity Ωy at the central plane y/W = 0 for all cases. The structure of Ωy is consistent with previous studies [28,80]. Furthermore, it closely resembles that of Δτrel (Fig. 15); the regions where τmta and τhom show significant differences correspond to the regions of high vorticity (i.e. inside the shear layer and along the downwind wall). As pointed out by Caton et al. [78] and Cui et al. [80], this is due to propagation of Kelvin-Helmholtz waves and the phenomenon is highly intermittent. The large Δτrel in this region suggests that the age spectrum can capture the intermittent behaviour of the flow. This effect is now quantified. The enhanced region extends to z / H ∼ 1/2 , z / H ∼ 1/5 and z / H ∼ 1/7 in cases U1, U2 and U3, respectively. For the non-uniform arrays, the high vorticity region extends beyond z / H2 ∼ 1/3. This is consistent with Fig. 7 which shows enhanced removal of pollutant in non-uniform arrays.

5.5. Effect of wind directions on age spectrum The direction of ambient wind has a significant impact on the mean tracer age and its statistical distribution. With an approaching wind parallel to the primary streets, the removal of pollutant mainly occurs across the roof top boundary (see Fig. 16). The upstream canyon units have distinct ventilation characteristics when compared to the downstream ones (see Fig. 9). With an approaching wind at an angle to the primary streets, however, ventilation in canyon and channel units has different characteristics. The canyon- or channel-averaged mean age for different wind directions is summarized in Table 5. For the canyon units, τmta decreases 11.5% when the wind direction changes from θ = 0° to 45° while the channel unit, τmta increases 20%. Here, θ represents the angle between the approaching wind to the primary streets. This implies that exposure of pollutants increases inside the channel units when the approaching wind is directed at an angle to the main streets. Fig. 18 displays the age spectrum of a channel or canyon unit for different wind directions. The distribution narrows as the wind changes direction from θ = 0° to 45°. The channel unit, however, exhibits the opposite trend. This supports the idea that lateral escape through the channel units increases with θ. To explain the effect of wind direction on the age spectra, the instantaneous z-vorticity at height z = 0.5H is displayed in Figure 19. The lateral vortices along the side walls propagate into the channel units as θ increases. By comparison with θ = 0° (Fig. 17), vortex shedding is no longer restricted to the canyon units, but now mixes the pollutants from the canyon units with those in the channel units. This is especially noticeable for θ = 45° (Fig. 19(b)). Thus, the mean pollutant age increases in the channel unit but decreases in the canyon units. Hence there is a more uniform spread of pollutants across the entire urban domain as compared to θ = 0°.

Fig. 16. Mean turbulent kinetic energy k and streamlines in the central x-z plane: (a) case U1, (b) case U2, (c) case U3, (d) case N1, and (e) case N2.

Table 3 Ratio of the mean tracer age to the average canyon circulation timescale for cases U1, U2 and U3.

τmta/ Tc

Case U1

Case U2

Case U3

0.44

0.36

0.29

Table 4 Ratio of the mean tracer age to the average canyon circulation timescale for cases N1 and N2.

We now explore the relationship between the mean tracer age and mean flow for the non-uniform arrays. Consistent with previous studies [11], the recirculation zone no longer exists; the flow has a mean downward motion in the canyon downstream of the shorter buildings and an upward motion upstream (see Fig. 16(d) and (e)). Timescale of the mean flow in the non-uniform canyon units may be estimated by

canyon 4 canyon 5

301

Case N1

Case N2

0.64 0.62

0.82 0.72

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Fig. 17. Mean y-vorticity Ωy in the central x-z plane: (a) case U1, (b) case U2, (c) case U3, (d) case N1, and (e) case N2.

Table 5 Canyon- or channel-averaged mean tracer age τmta idealized building array configuration case U1.

c

processes that affect the distribution of a passive tracer. Sensitivity of the age spectrum to different initial conditions was examined by considering source release locations at the ground level and inlet. For the inlet source, the origin of the approximately exponential tail shifts with the distance from the source to the receptor; hence the mean pollutant age is smaller for the ground-level source. The applicability of the mean tracer age to the analysis of urban ventilation was investigated through comparisons with the homogeneous emission method. Quantitative differences are found in regions where the flow is highly unsteady and intermittent. This is consistent with the time-averaged spanwise vorticity, thereby suggesting that the age spectrum is more effective for inhomogeneous flows. For LES, the computational time required by the age spectrum is not much more than the homogeneous emission method since both approaches require the solution of the advection diffusion equation. The main difference is that the age spectrum cannot be applied to steady flows, such as one simulated by RANS.

for different wind directions and

Wind direction, θ

Canyon unit

Channel unit

0-deg 30-deg 45-deg

223 s 211 s 200 s

174 s 205 s 210 s

6. Conclusion The age spectrum, or the probability distribution function of pollutant ages, was used to characterize ventilation in an inhomogeneous urban environment. Since it is obtained directly from the evolution equation of the passive scalar, it encapsulates the fluid dynamical 302

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Fig. 18. Effect of approaching wind direction on age spectrum Z y : (a) canyon unit and (b) channel unit. Pollutant source is released at the ground level.

entrainment is also 80% larger for non-uniform arrays. We have shown that the age spectrum and mean pollutant age are useful indicators for inhomogeneous urban domains. These indicators can be used to identify regions which are poorly ventilated and susceptible to high pollutant exposure. They should be beneficial for the development of effective methods of improving ventilation in real cities.

The quantification of ventilation, which refers to the removal of pollutants in the present context, represents the primary contribution of this work. Increasing the aspect ratio (building height/street width) from 1 to 4 reduces ventilation by approximately 67%. For non-uniform arrays, varying the building heights by 33% results in an improvement of about 30%. An analysis of the mean pollutant ages for inlet and ground-level sources demonstrates that the effect of fresh air

Fig. 19. Instantaneous z-vorticity Ωz in the x-y plane at z/W = 0.5: (a) 30°, and (b) 45°.

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Acknowledgments

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