Influence of Urban Morphologies on the Effective Mean Age of Air at Pedestrian Level and Mass Transport Within Urban Canopy Layer

Abstract

This study adapted the mean age of air, a time scale widely utilized in evaluating indoor ventilation, to assess the impact of building layouts on urban ventilation capacity. To distinguish it from its applications in enclosed indoor environments, the adapted index was termed the effective mean age of air (${\overline{\tau }}_E$). Based on an experimentally validated method, computational fluid dynamic (CFD) simulations were performed for parametric studies on four generic parameters that describe urban morphologies, including building height, building density, and variations in the heights or frontal areas of adjacent buildings. At the breathing level (z = 1.7 m), the results indicated three distinct distribution patterns of insufficiently ventilated areas: within recirculation zones behind buildings, in the downstream sections of the main road, or within recirculation zones near lateral facades. The spatial heterogeneity of ventilation capacity was emphasized through the statistical distributions of ${\overline{\tau }}_E$. In most cases, convective transport dominates the purging process for the whole canopy zone, while turbulent transport prevails for the pedestrian zone. Additionally, comparisons with a reference case simulating an open area highlighted the dual effects of buildings on urban ventilation, notably through the enhanced dilution promoted by the helical flows between buildings. This study also serves as a preliminary CFD practice utilizing ${\overline{\tau }}_E$ with the homogenous emission method, and demonstrates its capability for assessing urban ventilation potential in urban planning.

1. Introduction

Urbanization is an essential process of regional economic growth. Through the centralization of investment and emerging industries, cities provide improved public services and expanded opportunities for individual development, attracting inhabitants from surrounding areas [1]. While the concentration of industrials and the population enhances resource efficiency, the transformation of urban morphology, along with intensified human activities, presents significant challenges to human settlements.

To accommodate a thriving urban population within a limited urban space, cities are becoming denser while also expanding. Urban densification increases resistance to incoming ambient air, reducing the potential for the dilution and removal of contaminants, which further aggravates air pollution. Air pollution is one of the most pressing challenges, and continues to pose a severe threat to public health [2,3]. Nearly the entire global population breathes air that exceeds the unhealthy levels suggested by the World Health Organization (WHO) [3,4]. The dire state of air quality worldwide is a silent pandemic, with approximately 4.2 million premature deaths attributed to air pollution in 2019, increasing to 8.1 million in 2021 [5]. Therefore, controlling air pollution levels within favorable standards while maintaining high land-use efficiency is a crucial issue in contemporary urbanization. In addition to reducing emission as a direct approach, regional meteorological forces and urban layout designs have profound impacts on urban air quality [6]. Given the long-lasting nature of urban infrastructure, strategic urban planning is urgently needed for sustainable developments with favorable ventilation capacity, which requires thorough studies of the interactions between artificial structures and their surrounding environments.

The bottom section of the atmospheric boundary layer between the ground and the rooftops of buildings is defined as the urban canopy layer (UCL), where the majority of human activities take place [7]. Within the UCL, the variety of urban elements creates a great heterogeneity of microclimates. Therefore, aside from case studies of existing or planned urban regions that often reproduce the target area in detail [8,9,10], idealized models with generic building configurations are widely adopted to quantitatively examine the impacts of one or several urban parameters through controlled variable studies [11,12].

Flow conditions and pollutant dispersion in the UCL have been extensively studied through computational fluid dynamic (CFD) simulations, controlled laboratory experiments, and on-site field measurements [13]. Meteorological conditions are the fundamental forces of urban flows [8,9,14]. As for the artificial constructions within the UCL, the role of key metrics of urban morphology, such as the planar area index λ_p (also building density) and frontal area index λ_f, has been extensively investigated [8,15,16,17,18,19,20]. There are also many studies that focus on specific parameters, such as building height variations [21,22,23,24,25,26] and urban vegetation [27,28]. A variety of ventilation indices have been utilized to quantitatively describe the ventilation conditions and air quality within the UCL, comprising the air exchange rate, visitation frequency, residence time, purging flow rate, and mean age of air [29,30,31].

In this study, the mean age of air is adopted to assess the relation between ventilation capacity and the urban geometries of an idealized street block. Originally defined in building ventilation studies, the mean age of air (${\overline{\tau }}_P$) is a time scale that represents the statistically averaged travel time for air parcels from the entrance to an arbitrary position (P) within the control volume [32]. By regarding the entire or a target region within the UCL as an assembly of interconnected, roofless rooms, the mean age of air has been widely used in numerical studies to assess the ventilation efficiency, which is primarily attributed to the entrainment of ambient fresh air [19,33,34]. However, compared to the ventilation for an enclosed room, boundaries of the UCL exhibit a bi-directional mass transport driven by both convection and turbulence, leading to the re-entry of polluted air into the control volume [7,13,35]. Therefore, the index adapted in this study was termed the effective mean age of air (${\overline{\tau }}_E$) in order to clarify its distinction from applications in indoor scenarios. This study is part of a research project with the goal of developing a preliminary method for assessing the urban ventilation capacity of a given region in the planning stage. In this context, when practical emission data are likely to be unavailable, the mean age of air is a more feasible metric due to its independence from the release rate [29,32]. Furthermore, indoor air quality is significantly influenced by outdoor conditions in surrounding street networks, particularly for naturally ventilated buildings [36]. As indoor and outdoor air are essentially coupled, it is beneficial to evaluate ventilation conditions using a consistent metric in both contexts, which facilitates communication between professionals and contributes to a coherent system view of urban ventilation.

CFD approaches are adopted in this study due to their flexibility for conducting parametric studies. To derive the mean age of air within the UCL, it is vital to define the boundary of the focused control volume, through which the fresh air enters. Previous numerical studies using the mean age of air defined the entire UCL as a control volume, labeled by a hypothetic homogeneous emission [19,34,37]. This approach reduces deviations introduced by re-entrained polluted air, and it may also obscure dilution processes within the UCL and the spatial heterogeneity of the local ventilation capacity, especially when urban emissions are mostly attributed to vehicles at a pedestrian level. Therefore, the pedestrian level—defined as the space from z = 0 m to 2 m across the built area—is instead specified as the control volume in this study. Additionally, mass transport attributed to the mean flow and turbulence across different interfaces within the UCL is quantified according to [38], which further examines the interior mass transport.

Furthermore, the relative enhancement or deterioration of the ventilation capacity due to the presence of buildings is assessed by calculating the normalized age of air. Instead of normalizing the effective age of air by a reference flow rate defined with the area of the windward entry of built area, as introduced by [34], a reference case representing an open area without buildings was simulated. Adapted from the definition of air delay by Antoniou et al. [37], this method also accounts for the streamwise accumulation of upstream pollutants. The contours of this non-dimensional index provide more intuitive references for urban planning.

2. Methodology

This section introduces the key quantities for quantitative analysis, as well as the validation and configurations for CFD simulations. The effective mean age of air and the normalized mass transport rate are defined in Section 2.1 and Section 2.2, respectively. Section 2.3 presents the numerical methods. The wind tunnel measurement by Brown et al. [39] was adopted to evaluate the performance of the chosen numerical method—the RANS method with the standard k-ε model for turbulence and standard wall function—and served as the basic urban layout for subsequent case designs. Consequently, various modifications were made to the basic urban model to construct four groups of cases, each focused on a generic urban morphology factor.

2.1. Effective Mean Age of Air

With homogenous emission within the room, which labels the air by tracer gas, the relation between the time-averaged concentration of tracer gas ($\overline{c}$) and ${\overline{\tau }}_P$ was mathematically derived by Etheridge and Sandberg [32]. By regarding either the whole or an interested part of the UCL as a control volume, the effective mean age of air (${\overline{\tau }}_E$) can be calculated by a similar method. In numerical simulations, the air within specified zones can be marked by a hypothetical passive emission without initial momentum. As introduced in CFD studies by Hang et al. [34], the local effective mean age of air is then calculated as in Equation (1), as follows:

$${\overline{\tau }}_E={\overline{c}}_p/\dot{m}$$

(1)

where ${\overline{c}}_p$ is the local concentration of a passive tracer gas (kg/m³), and $\dot{m}$ is the uniform release rate (kg/(m³s)). Insufficient air exchange leads to a longer time for the external air to reach an arbitrary location, resulting in a higher ${\overline{\tau }}_E$ at that position, which is independent of the hypothetical emission rate according to the algorithm.

2.2. Normalized Mass Transport Rate

As introduced in [38], based on results from steady-state simulations utilizing RANS models, the convective mass flux by mean flow can be defined as follows:

$$Q_{m, i}=\overline{u_i}·\overline{c}$$

(2)

while the mass flux transported by turbulence is derived based on the following gradient diffusion hypothesis:

$$Q_{t, i}=-K_c{\displaystyle \frac{\partial \overline{c}}{\partial x_i}}$$

(3)

Here, ${\overline{u}}_i$ (i = 1, 2, or 3) represents the time-averaged velocity components in the streamwise, spanwise, and vertical directions, respectively. The variable $\overline{c}$ denotes the average concentration (kg/m³). K_c is the turbulent eddy diffusivity of the species, defined as $K_c=v_t/S_{ct}$, where the eddy viscosity ${\nu }_t=C_{\mu }{\displaystyle \frac{k^2}{\epsilon }}$ ($C_{\mu }=0.09$, a model constant). And Sc_t is the turbulent Schmidt number, set to be 0.7 according to the literature modeling the pollutant dispersion in a similar context [9,12,17,28].

The mass transfer through a surface is calculated by the surface integral of the perpendicular mass flux. For comparisons across cases, the flux integrals are normalized by the total volumetric release rate of the target zone ($\dot{M}$ [kg/s]), as follows:

$$F_{m, i}=\int Q_{m, i}dA/\dot{M}$$

(4)

$$F_{t, i}=\int Q_{t, i}dA/\dot{M}$$

(5)

$F_{m, i}$ and $F_{t, i}$ are referred to as the normalized convective mass transport rate (F_m,i) and normalized turbulent mass transport rate (F_t,i), respectively [22]. As molecular diffusion is negligible, the sum of mass transport across each interface in all three directions should meet the mass conservation. The discussion in Section 3.2.2 is from the perspective of the volumetric mass balance; therefore, the subscript i indicating the direction is omitted hereafter.

2.3. CFD Setup and Case Description

2.3.1. Turbulence Model for Urban Ventilation Modeling

CFD simulations have been widely applied in the modeling of flow and thermal conditions within the UCL in past decades [11,40]. Among popular numerical methods, the Large Eddy Simulation (LES) has been known to perform better in predicting turbulence than Reynolds-Averaged Navier–Stokes (RANS) approaches. Nevertheless, challenges still persist in the widespread application of the LES, typically including the high requirement of computational resources and difficulties in specifying appropriate time-dependent boundary conditions [41,42,43,44]. Despite their deficiencies in predicting detailed turbulence features, steady RANS models have become the most popular CFD approaches for modeling urban flows, and the standard k-ε model is one of the most widely adopted two-equation turbulence model [11,17,45]. Since transient features were not the focus in this study, the standard k-ε model was adopted for the simulations due to its adequate accuracy, which has been validated by experimental measurements, and its reliability in predicting time-averaged features. Additionally, as parametric studies require an adequate set of cases to quantitatively evaluate the impact of a specific factor, it is more practical to adopt RANS approaches considering the overall computational costs.

The commercial software Ansys FLUENT (2022 R1) was utilized for the simulations. The steady-state incompressible and isothermal flows in full-scale urban models under neutral atmospheric conditions were modeled using the standard k-ε model and standard wall function. The governing equations for time-averaged velocity, turbulence, and species transport are provided in Chapters 1.2, 4.3, and 7.1 of the Fluent Theory Guide [46]. All governing equations were discretized with the second-order upwind scheme. The SIMPLE algorithm was employed to couple pressure and velocity. The under-relaxation factors were set as 0.3, 0.7, 0.8, and 0.8 for the pressure term, momentum term, and k and ε, respectively. The simulation would be stopped after 20,000 iterations, ensuring that the magnitudes of residuals for typical variables were below 10⁻⁷.

2.3.2. Validation for CFD Methods

The accuracy of the chosen CFD approach, using standard k-ε models and standard wall function, was evaluated against wind tunnel measurement data from Brown et al. [39]. The mesh independence of the simulation results was also checked. The model setting in the original wind tunnel experiment is illustrated in Figure 1, which is an idealized urban model comprising 7 rows and 11 columns of equally spaced cubes, each with 15 cm sides and a spacing ratio of 1 (H/W = 1). The incoming flow was parallel to the main streets, as denoted by the x-axis direction. When the incoming flow velocity at the model height (referred to as U₀ hereafter) equals 3 m/s, the wind tunnel models possess a characteristic Reynolds number (Re) = 30,625, calculated by the following definition equation:

$$Re={\displaystyle \frac{U_0\cdot H}{\nu }}$$

(6)

in which the H = characteristic dimension here equals the building height, and ν = kinematic viscosity of air.

For numerical validation cases, models were scaled up by a ratio of 200:1 relative to the original wind tunnel models. Since the widely adopted criterion of Re independence (Re > 11,000) was met [47], the CFD validations conducted at full scale would reproduce the same flow filed as the downscaled wind tunnel experiment, as the flow pattern is fully developed at both scales. Chew et al. 2018 [47] also suggested that high aspect ratios increase the threshold Re for the modeled flows to gain dynamic similarity with the realistic flow at a larger scale. Therefore, scaled-up models are more efficient when considering dense urban layouts.

Figure 2 illustrates the calculation domain with model details, including half of the center model column and half of the adjacent passage. Consequently, the lateral boundaries were designated as symmetry boundaries. As the original experimental model has a wide-spanned, axis-symmetrical layout (Figure 1), the flow conditions within the middle part are away from disturbances at the lateral fringes of the model cluster, which has been validated by previous numerical studies employing similar calculation domains [21,39,48,49,50,51,52]. In addition, the top boundary of the calculation domain was also set as a symmetry boundary. The domain outlet was specified as outflow with zero normal gradient. A zero normal flux condition was set for all solid non-slip walls, including the floor. The size of the calculation domain was defined according to the AIJ CFD guideline for urban modeling [53].

A power-law velocity profile defined as Equation (7) was adopted for the velocity inlet, representing a neutral atmospheric boundary layer with an equivalent roughness height z₀ = 0.1 m [45]. The turbulent kinetic energy k(z) and its dissipation rate ε(z) are accordingly calculated by Equations (8) and (9) [39,48,54].

$$U\left(z\right)=U_0{\left({\displaystyle \frac{z}{H}}\right)}^{0.16}$$

(7)

$$k\left(z\right)={\displaystyle \frac{u_{\ast }^2}{\sqrt{C_{\mu }}}}$$

(8)

$$\epsilon \left(z\right)={\displaystyle \frac{C_{\mu }^{\frac{3}{4}}{k}^{\frac{3}{2}}}{{\kappa }_{v}z}}$$

(9)

In Equations (7)–(9), U₀ is the reference velocity magnitude of the approaching free flow at model height (z = H), which is 3 m/s in the wind tunnel experiments by Brown et al. [39]; $u_{\ast }$, the friction velocity, was assessed to be 0.24 m/s in the experiment [39,54]; $C_{\mu }$ is a model constant, which equals 0.09; and ${\kappa }_v$ is the von Karman’s constant and equals 0.41 [46].

The positive direction of the x, y, and z axes was defined as the streamwise, spanwise, and vertical direction, respectively. The coordinate x/H = 0 denotes the windward entry of the built area, and y/H = 0 goes along the center axis of the target column.

Three different grid arrangements were tested for grid independence of the simulation results (Figure 3), with a grid expanding ratio between 1.12 and 1.2. There were at least four layers of grids adopted at the pedestrian level (z = 0 to 2 m), which meets the grid requirement by the CFD guidelines [53,55]. As the lower limit of the normalized distance of wall surfaces (i.e., $y^{+}=y{u}_{\tau }/\nu$) in the full-scale cases with height Re was around 50, standard wall function was applied [45]. Detailed information for validation cases is summarized in

Table 1. Summary of cases for CFD method validation and mesh independence test.

Case Name	Turbulence Model	Minimum Grid Spacing	Cell Number
[c, std]	Standard k-ε model	0.1 m	1,870,197
[f, std]	Standard k-ε model	0.4 m	1,005,497
[m, std]	Standard k-ε model	0.2 m	1,397,744
[m, rng]	RNG k-ε model	0.2 m	1,397,744

. The RNG k-ε model was used in one case for comparison.

The comparisons between CFD results with wind tunnel data at sampling positions V_i (i = 1–5, as denoted in Figure 1) are illustrated in Figure 4. The vertical profiles of the time-averaged streamwise velocity $\overline{u}\left(z\right)$, the vertical velocity $\overline{w}\left(z\right)$, and the turbulence kinetic energy k(z) with all three grid arrangements and both turbulence models are presented. As shown in Figure 4, the standard k-ε model outperformed the RNG k-ε model in all quantities compared. The results from the three cases applying the standard k-ε model with different grid arrangements show minimal difference. Additionally, the results with the medium gird arrangement exhibit very little difference compared to those obtained using fine grids. Therefore, the standard k-ε model with medium grid spacing is employed in subsequent case studies to optimize computational efficiency while ensuring sufficient numerical accuracy.

2.3.3. Case Descriptions for Parametric Studies

Based on the idealized urban model validated above, various modifications on the model geometries were implemented to study the influences of interested morphology characteristics on the urban ventilation capacity. The model layout in the validation case was adopted as the base case, while the computational domain is expanded following guidelines [53,55] to accommodate various adaptations on models for parametric studies, as illustrated in Figure 5. The model height of 30 m in the validation case is taken as the reference building height, denoted as H₀.

Four generic parameters describing urban morphologies—building height, building density, and variations in the heights or frontal areas of adjacent buildings—were studied by four groups of simulation cases. The model geometry and arrangement in each case were modified according to the focused factor, as illustrated in Figure 6. Group A investigated the influence of building heights, with uniform model heights ranging from 10 m to 60 m across cases (Figure 6a). Group B maintained the building height as 30 m, while varying the spacings between buildings to achieve the desired building densities in each case. The cases in group C feature height variations between adjacent building rows, with the 2nd, 4th, and 6th rows at a distinct height H₁ rather than H₀, which ranges from 5 to 25 m, increasing by 5 m between cases. In group D, the widths B of adjacent buildings vary while maintaining a constant sum of frontal areas of any two adjacent models as 2 × B × H₀ m². The alteration in width B varies from 10% to 50% of H₀, with a consistent increment of 10% across the cases. In addition, the calculation domain for group D was extended in the spanwise direction to ensure geometrical symmetry across the lateral boundaries, as illustrated in the top view in Figure 6d.

The boundary conditions and coordinate system are identical with the validation case. The impact of varying incoming flow velocities is also investigated with cases in group A, for which the U₀ of the inlet velocity profile is adjusted to 1, 3, 5 m/s, respectively.

Table 2. Case descriptions and configuration summary.

Group	Case Naming	Configuration	Value Range of the Specified Variables
A	[H-H, SP1, u]	Models in equal height with identical spacing (W = H0), tested with 3 different incoming flow velocities (Figure 6a).	H = 10, 20, 30, 40, 50, 60 (m) U0 = 1, 3, 5 (m/s)
B	[30-30, SPr, 3]	All cases maintain a consistent model height of 30 m (H0) and a constant total length of the built area (Ls = 390 m), while varying in spacings between (Figure 6b).	r = 0.5, 0.7, 1, 2, 3, represent the ratio between spacing W and reference building height H0, i.e., W/H0 = r.
C	[30-H1, SP1, 3]	Same layout as the base case, while models in the 2nd, 4th, and 6th row have a different height noted as H1 (Figure 6c).	H1 = 5, 10, 15, 20, 25 (m)
D	[30-30, SFp, 3]	The width of the frontal facade alternatively increases or decreases by p% of H0, while keeping a constant total frontal area for each pair of adjacent models (Figure 6d).	p = 10, 20, 30, 40, 50 (%)

[30-30, SP1, 3] is the base case described in Figure 5. For brevity, the number indicating incoming flow velocity is omitted in discussions when it is irrelevant.

summarizes the naming rules and corresponding model adjustments for each group.

Different partitions are specified as shown in Figure 7. The pedestrian zone, highlighted in red in Figure 7a, is defined as the region from z = 0 m to 2 m, covering the entire built area from the first to the last building row (i.e., 0 ≤ x ≤ L_s, 0 ≤ y ≤ W_domain, and 0 ≤ z ≤ 2 m). The canopy zone is defined between the roof level of the highest building in the calculation domain and the ground (Figure 7b). Both the pedestrian zone and the canopy zone have an inlet interface at the upstream entrance, and an outlet at the downstream exit of the built area between their respective roof and ground levels. Furthermore, the spaces between the building rows within the pedestrian zone are designated as Crossroads (Figure 7c), labeled as Cn, where n denotes behind which building row the crossroad is located. Accordingly, the street parallel to the incoming wind is named as the Main Road (Figure 7d). In addition, the breathing level is specified at z = 1.7 m across the built area, where flow conditions and distributions of ${\overline{\tau }}_E$ will be stressed.

As mentioned in Section 2.1, the effective mean age of air (${\overline{\tau }}_E$) was derived using the homogenous emission method to label the air within the focused region [19,34]. In this study, the air within the pedestrian zone is labeled by the uniform homogeneous emission of carbon monoxide (CO), with a constant release rate of 10⁻⁵ kg/(m³s). Therefore, the air outside the pedestrian zone is regarded as fresh air.

3. Results and Discussion

3.1. General Flow Conditions

Typical features of flow conditions are discussed first. The contour plots of velocity magnitude with 2D streamlines at the breathing level (z = 1.7 m) and 3D streamlines within the built area in representative cases are presented in Figure 8 and Figure 9. Complete figures for all cases are provided in the attachment to manage the length of the paper.

Within building models of uniform height in each instance (group A, Figure 8a–c), the flow velocity along the main road initially decreases as the building height increases from 10 m to 30 m, but then increases when the height further increases to 60 m, while the velocity magnitude behind buildings changes inversely. In contrast, the flow pattern at breathing level changes monotonically as the building height increases; the vortices confined behind low buildings by the strong streamwise flow diminish, while outward flows emerge and become dominant as building height increases. The 3D streamlines (Figure 9a–c) visualize the transformation of flow patterns behind buildings with increasing building height. Notably, there are two typical flow patterns: vertical vortices behind medium-height buildings (Figure 9b) and helical flows featuring downwash behind tall buildings (Figure 9c).

The spacing between buildings dramatically affects the flow rate penetrating into the UCL, as indicated by the contrasting velocity magnitudes shown in Figure 8d for case [30-30, SP0.5] and Figure 8e for case [30-30, SP3]. Figure 9e denotes a lifting trend of the flow along the main road with extremely heigh building density (SP3), which enables the formation of recirculation zones near the lateral facades by the outflows from the downwash behind buildings, corresponding to the small vortices shown in Figure 8e.

In group C, when building models of two different heights are placed alternatively along the streamwise direction, the downwind wake behind a tall building can reach the next tall building if the intermediate building is sufficiently short, as depicted by the 3D streamlines in Figure 9f. This explains the similarities in the flow patterns illustrated in Figure 8d,f, as the obstruction caused by the shorter building is limited to the near-ground level. When the height difference decreases, the flow behind each building becomes more independent, approaching the condition of the base case shown in Figure 8a.

For the cases in group D, with the same total frontal area for each row, the alternating frontal areas of the two buildings on the left and right side bend the inflow into a curved shape throughout the built area (Figure 8g), with the curvature proportional to the difference between the frontal areas. Notably, as case [30-30, SF50] shows in Figure 8g, significant recirculating flows appear behind the wider buildings due to their considerable blockage in the streamwise direction.

To conclude, the relative strength between the flow along the main road and the flow from the crossroads is critical for the direction of pollutant transport and locations of accumulation sites. Accordingly, two typical flow patterns at the breathing level can be summarized: (1) the streamwise main road flow prevails, accompanied by confined recirculation flows behind tall buildings (e.g., Figure 8a,d,f); (2) the main road flow diminishes, allowing outward flows generated by downwash in the crossroads to converge with it (e.g., Figure 8b,c,e). This feature is also substantiated by the sole vortex in Figure 8c, located behind the first building, where the streamwise flow at the entrance of the main road is relatively stronger than in the areas further downstream.

3.2. Impact of Building Configurations on Local Ventilation Capacity and Mass Transport

3.2.1. Local Ventilation Capacity Described by Effective Mean Age of Air

According to the definition of effective mean age of air (${\overline{\tau }}_E$) in Section 2.1, it is adequate to describe the local ventilation capacity regardless of the arbitrary release rate used in the simulation. A higher ${\overline{\tau }}_E$ value signifies reduced ventilation efficiency, and an increased risk of pollution accumulation at that location.

Figure 10 presents the distribution of ${\overline{\tau }}_E$ at the same breathing level for those representative cases. The locations of high ${\overline{\tau }}_E$ areas are highly consistent with the recirculation zones, as shown in the velocity contour in Figure 8a,d,f,g. Meanwhile, in cases with a weak main road flow and relatively stronger outward flow from the crossroads, the downstream part of the main road shows a higher ${\overline{\tau }}_E$ due to accumulation (Figure 10b,c,e). The distributions of high ${\overline{\tau }}_E$ areas across cases suggests three primary locations of poorly ventilated sites: within the vortices behind buildings (e.g., Figure 10a,d,f,g), the downstream section of the main road (Figure 10b,c), or within the recirculation zones beside the lateral facades of buildings (Figure 10e).

Varied distribution patterns of regions with a high ${\overline{\tau }}_E$ at the breathing level emphasize the spatial heterogeneity of ventilation conditions with different building layouts. However, these extreme values can easily be obscured by averaging, which indicates high risk of exposure to high pollution from a ventilation perspective. Therefore, detailed statistical analyses of ${\overline{\tau }}_E$ values at the focused breathing level were conducted. Discrete ${\overline{\tau }}_E$ values extracted from cell centers were interpolated into equally spaced matrices to eliminate the influence of area weighting and discrepancies between meshes with different geometries. The statistical characteristics of the ${\overline{\tau }}_E$ values at the breathing level within the built area for each case are presented as a box plot in Figure 11. For the ${\overline{\tau }}_E$ values at the breathing level section, when the maximum value exceeds the whisker limit, it demonstrates the presence of extreme ${\overline{\tau }}_E$ outliers, with the distance in between indicating the distinctiveness of the high ${\overline{\tau }}_E$ center within the field. Moreover, the gap between the 95th percentile and median values is proportional to the area of high ${\overline{\tau }}_E$. These statistical assessments align closely with the features observed in ${\overline{\tau }}_E$ contours, as shown in Figure 10 for representative cases, and additional contours for other cases are provided in Appendix A.

The average ${\overline{\tau }}_E$ values of the entire breathing level section for the cases in groups A, C, and D are close, ranging from 30 s to 40 s (Figure 11a,c,d), while the ${\overline{\tau }}_E$ values corresponding to the 95th percentile and the maximum vary significantly across cases. In group A (Figure 11a), cases [40-40, SP1] and [50-50, SP1] exhibit higher maximums and averages, rather than the case with the tallest buildings. This fact is attributed to the combined effect of a relatively weaker streamwise main road flow and moderate convergence from the crossroad flows, as discussed in Section 3.1. Additionally, the box plot for case [10-10, SP1] implies distinct ${\overline{\tau }}_E$ outliers, aligning with the presence of high-${\overline{\tau }}_E$ centers within the recirculation zones behind buildings (Figure 8a and Figure 10a), which diminish when the building height increases to 20 m. Meanwhile, for group B, as shown in Figure 11b, the increasing building density significantly raises both the average value and the proportion of outliers, quantitatively depicting the emergence of larger areas with a high ${\overline{\tau }}_E$ when W < H₀ (i.e., SP > 1). In group C, the intermediate building obstructs the mixing between the two crossroads as its height increases. Meanwhile, the flow intensity along the main road becomes weaker when the average building height increases, which benefits the purging of the crossroads by the outward flow. Both mechanisms concurrently modify mass transport. Consequently, cases [30-20, SP1] and [30-15, SP1] suffer both the accumulation within the recirculation behind the tall buildings and deteriorated mixing between crossroads, which corresponds to elevated maximum ${\overline{\tau }}_E$ values in Figure 11c. Nevertheless, in group D, the increasing difference in the frontal area of the two adjacent buildings consistently deteriorates ventilation conditions at the breathing level, attributing to the enhanced recirculation zones behind wider buildings (Figure 8g and Figure 10g), leading to severe pollutant accumulation and extreme ${\overline{\tau }}_E$ outliers, as shown in Figure 11d.

As discussed in previous subsections, the relative dominance between the main road flow and crossroad flow dramatically influences the transport routes and accumulation sites. Within an orthogonal street network, as constructed in this study, pollutant dispersion can accordingly be decomposed into two transport directions: streamwise and spanwise. Accordingly, the ventilation conditions along these two directions at the breathing level are outlined, respectively.

The spanwise (y direction) integrations of ${\overline{\tau }}_E$ along the main road are presented in Figure 12. Across all cases, the dips shown in the line charts mostly contributed to the outward crossroad flows, attributed to the downwash close to the windward facade. There are also several noteworthy results within each group. Figure 12a shows more accumulation in the downstream part of the main road for cases [40-40, SP1] and [50-50, SP1], which is consistent with the result shown in Figure 11a. With high building densities, the small recirculation zones have extremely high ${\overline{\tau }}_E$ values, attached to the lateral facades facing the main road in cases [30-30, SP2] and [30-30, SP3], leading to the surge in ${\overline{\tau }}_E$, as shown in Figure 12b. Additionally, the peak value at the end of the road in case [30-30, SP3] stems from the backward flow branch of the vortex after the whole building row, which can be found in Figure 9e. Notably, with variation in either building heights or the frontal area, the ventilation conditions on the main road are improved compared to the base case (Figure 12c,d), as the high-value centers are located at the crossroads behind the buildings, according to the contour plots in Figure 10f,g.

Graphs of the average ${\overline{\tau }}_E$ for each crossroad at the breathing level for all cases are exhibited in Figure 13, indicating the potential of spanwise transport in the built area. While the results presented align with previous discussions, some points need further clarification. Since the crossroads include crossings with the main road (Figure 7d), a higher ${\overline{\tau }}_E$ at the main road could also raise the average ${\overline{\tau }}_E$ values of the crossroads. Therefore, the plot of case [60-60, SP1] shows a moderate increasing rate of the average ${\overline{\tau }}_E$, as the streamwise flow and transport rate are weaker. The drop in the average ${\overline{\tau }}_E$ of the crossroads close to the outlet boundary of the built area in cases [30-30, SP2] and [30-30, SP3] attributes to the enhanced outward flow from the building intervals to the main road, as inferred from Figure 8e.

3.2.2. Quantifying the Contribution of Convective and Turbulent Transport to the Total Removal for the Built Area

How different building layouts influence the dominant mass transport mechanisms at various positions within a built environment is also a key issue in urban planning and renovation. In this subsection, the relative predominance of convective and turbulent transport to the overall pollutant removal of a control volume is quantitatively analyzed. The three representative sections shown in Figure 7a–c are utilized to highlight regional features.

Detailed quantitative results for the base case [30-30, SP1, 3] are presented for reference.

Table 3. Normalized mass transport rates through interfaces of canopy zone and pedestrian zone of the base case [30-30, SP1, 3].

	Canopy			Pedestrian
	Fm	Ft	Fsum	Fm	Ft	Fsum
Inlet	4.84 × 10−4	−7.15 × 10−4	−1.01 × 100	2.36 × 10−4	−2.43 × 10−4	−9.89 × 10−1
Outlet	−6.04 × 10−1	−3.15 × 10−3		−2.55 × 10−2	−8.55 × 10−4
Roof	−1.45 × 10−2	−3.85 × 10−1		−2.92 × 10−1	−6.70 × 10−1

presents the sum of the normalized convective and turbulent transport rate (F_sum= F_m + F_t) through every interface of the canopy and the pedestrian zone, and

Table 4. Contribution fractions of convective and turbulent transport to the total removal in each segment of the base case [30-30, SP1, 3].

	C1	C2	C3	C4	C5	C6	Canopy	Pedestrian
Fm/Fsum	34%	37%	41%	45%	48%	52%	61%	32%
Ft/Fsum	67%	63%	59%	55%	52%	48%	39%	68%
Fm/Ft	0.50	0.58	0.70	0.82	0.93	1.07	1.59	0.47

lists the ratios between F_m and F_t for each section. According to the definitions of F_m and F_t by Equations (4) and (5), the total F_sum across all interfaces for a given control volume containing an emission source should equal −1 to satisfy the mass conservation, which is met for both canopy and pedestrian zones, as shown in Table 3. The minor deviations are mainly attributed to the mesh separation and the locations of interfaces defined for data processing.

The contribution ratios between two transport mechanisms for crossroads are significantly influenced by their distances to the upwind inlet of the built area (Table 4). Closer to the upstream inlet, turbulent transport tends to be dominant. Meanwhile, convective transport is strengthened in the downstream sections, contributing to around half of the total removal. For the entire pedestrian zone, turbulent transport contributes more than twice as much as convective transport. Conversely, for the canopy zone, convective transport is more prominent, primarily attributed to the mean flow through the downstream outlet of the built area.

Mass transport through the roof interfaces of both canopy and pedestrian zones is prioritized for discussion, as it has the largest area compared to inlet and outlet interfaces. For the pedestrian zone, the total transport across the roof interface dominates in all cases, contributing over 90% to the total removal, due to its significantly larger area and relatively weaker horizontal flow near ground level. In contrast, total transport through the outlet interface is more decisive for the canopy zone, while the vertical concentration gradient at the canopy roof is significantly decreased due to the distance from the emission. This fact is supported by the distinct drop in the percentage of total mass transport through the canopy roof as the building height increases, which simultaneously enlarges the outlet interface, as shown by the results of group A in

Table 5. Contribution fractions of mass transport through the roof interface of the canopy zone and the pedestrian zone in each case.

	[10-10, SP1, 3]	[20-20, SP1, 3]	[30-30, SP1, 3]	[40-40, SP1, 3]	[50-50, SP1, 3]	[60-60, SP1, 3]
Canopy roof	62%	41%	40%	34%	19%	6%
Pedestrian roof	91%	93%	97%	99%	99%	97%
	[30-30, SP0.5, 3]	[30-30, SP0.7, 3]	[30-30, SP1, 3]	[30-30, SP2, 3]	[30-30, SP3, 3]
Canopy roof	26%	29%	40%	69%	73%
Pedestrian roof	94%	95%	97%	100%	100%
	[30-30, SP1, 3]	[30-25, SP1, 3]	[30-20, SP1, 3]	[30-15, SP1, 3]	[30-10, SP1, 3]	[30-05, SP1, 3]
Canopy roof	40%	40%	43%	45%	46%	46%
Pedestrian roof	97%	97%	97%	96%	96%	96%
	[30-30, SF0, 3]	[30-30, SF10, 3]	[30-30, SF20, 3]	[30-30, SF30, 3]	[30-30, SF40, 3]	[30-30, SF50, 3]
Canopy roof	40%	40%	41%	43%	46%	51%
Pedestrian roof	97%	97%	97%	97%	97%	97%

. The balance between the mean flow and vertical transportation is also evidenced by the results from group B. With lower building densities and consequently less resistance to the incoming wind, the vertical transport across the canopy roof is lower than that of the base case (cases [30-30, SP0.5] and [30-30, SP0.7]), and conversely, higher building densities result in greater contribution from the roof interface (cases [30-30, SP2] and [30-30, SP3]). Moreover, variations in building heights (group C) and the frontal area (group D) both slightly enhance vertical transport through the canopy roof, as disturbances by the variations in building geometries increase the potential for vertical air movement.

Focusing on the entire canopy zone, the relative dominance of turbulent versus convective transport is further analyzed. Figure 14 quantitatively presents the contribution fractions of convective and turbulent transport for all cases, categorized by groups. As discussed above, the increasing building height in group A enlarges the area of outlet interface and consequently amplifies the contribution of mass transport through it, where the convective transport dominates (Figure 14a). In group B (Figure 14b), the inverse relationship between the F_m and aspect ratio is mainly attributed to the reduction in the convective flow rate through the built area as the building density increases. For groups C and D (Figure 14c,d), the disturbances in the flow field caused by the variations in building geometries promote turbulence, which in turn augment F_t.

Figure 15 presents the contribution fractions of convective and turbulent transport for the pedestrian zone, followed by

Table 6. F_m/F_t ratios for canopy zones and pedestrian zones.

	[10-10, SP1, 3]	[20-20, SP1, 3]	[30-30, SP1, 3]	[40-40, SP1, 3]	[50-50, SP1, 3]	[60-60, SP1, 3]
Canopy	0.55	1.26	1.59	2.96	8.77	31.52
Pedestrian	0.23	0.36	0.47	0.50	0.36	0.25
	[30-30, SP0.5, 3]	[30-30, SP0.7, 3]	[30-30, SP1, 3]	[30-30, SP2, 3]	[30-30, SP3, 3]
Canopy	2.67	2.36	1.59	0.77	0.53
Pedestrian	0.22	0.32	0.47	1.38	1.37
	[30-30, SP1, 3]	[30-25, SP1, 3]	[30-20, SP1, 3]	[30-15, SP1, 3]	[30-10, SP1, 3]	[30-05, SP1, 3]
Canopy	1.59	1.67	1.51	1.40	1.37	1.36
Pedestrian	0.47	0.50	0.48	0.41	0.33	0.26
	[30-30, SF0, 3]	[30-30, SF10, 3]	[30-30, SF20, 3]	[30-30, SF30, 3]	[30-30, SF40, 3]	[30-30, SF50, 3]
Canopy	1.59	1.58	1.57	1.51	1.39	1.27
Pedestrian	0.47	0.47	0.47	0.45	0.37	0.33

summarizing the ratios of F_m/F_t for both canopy and pedestrian zones across all cases. In group A (Figure 15a), the turbulent transport consistently dominates, while the fractions of both F_m and F_t show a U-shaped trend with increasing building height. As discussed in Section 3.2.1, cases [40-40, SP1] and [50-50, SP1] feature the highest maximum and a large portion of extremes of ${\overline{\tau }}_E$ values among all cases in group A at the pedestrian level. This fact indicates a higher overall concentration within the pedestrian zone for these two cases, which relatively enhances convective transport, according to the definition of F_m by Equation (4). In group B (Figure 15b), the relative magnitudes of F_m and F_t in relation to the building density present an inverse relationship compared to that for the canopy zone shown in Figure 14b. As illustrated by the 3D streamlines in Figure 9e for case [30-30, SP3], with narrow spacings, the streamwise flow ascends along the main road, accompanied by active upward flows from the near-ground level. In addition, a higher building density, with more buildings in the built area, leads to an increase in helical flows within the building intervals. Both facts contribute to stronger vertical flows and significantly promote convective transport through the pedestrian roof for cases [30-30, SP2] and [30-30, SP3] (Table 6). In groups C and D (Figure 15c,d), the variations in the geometry of adjacent buildings both slightly amplified the turbulent transport for the pedestrian zone, which is similar to the results of the canopy zone (Figure 14c,d).

3.2.3. Normalized Effective Mean Age of Air

A reference case without any building model was created, with a uniform emission covering the whole built area, ranging from z = 0 to 2 m, consistent with other cases. Maintaining the same calculation domain and boundary conditions, this reference case serves as a general baseline, providing the background field of ${\overline{\tau }}_E$ of an idealized open area. The reference case was simulated with three different incoming flow conditions, with U₀ as 1, 3, and 5 m/s separately, same for the base model. The ${\overline{\tau }}_E$ contour at the breathing level for the reference case with U₀ = 3 m/s is provided in Figure 16a. The other two incoming flow velocities did not alter the distribution pattern but only affected the magnitude of ${\overline{\tau }}_E$, so their results are omitted for brevity. For clarity, ${\overline{\tau }}_E$ for the reference case is noted as ${\overline{\tau }}_0$ instead hereafter.

The discrete ${\overline{\tau }}_E$ values at the breathing level of the base model are normalized by the corresponding ${\overline{\tau }}_0$ at the same coordinate. Contours of the normalized mean age of air are presented in Figure 16b. Values exceeding 2 are left blank in the contours to improve the readability of the color scales. Within the contours, a value of 1 indicates that ${\overline{\tau }}_E$ at an arbitrary position equals the corresponding ${\overline{\tau }}_0$ in the reference case, signifying that the impacts from the buildings are neutral at this position. Subsequently, a value of ${\overline{\tau }}_E/{\overline{\tau }}_0$ higher than 1 suggests a deteriorated ventilation condition, while a value of less than 1 indicates even improved ventilation potential compared to the same location in the open area.

As illustrated in Figure 16b, with this generic urban form in case [30-30, SP1], the presence of buildings deteriorates the ventilation condition along the main road. In contrast, between the buildings, the downwash from the helical flows enhances ventilation capacity compared to the reference case. These facts provide a different perspective from the common belief that buildings always worsen air quality. The results indicate that the influence of artificial structures has significant spatial heterogeneity, with both positive and negative effects. Additionally, the high consistency of the results with different incoming flow velocities demonstrates the capability of normalized ${\overline{\tau }}_E$ as a tool for preliminary evaluations of ventilation capacity or potential in the urban design. With a specified building layout and prevailing wind direction, it provides sufficient details on local purging capacity and resilience under assumed pollutant scenarios, without the need for detailed meteorological boundary conditions or real emission data.

More detailed parametric studies are required to establish clearer relationships between the normalized ${\overline{\tau }}_E$ and various urban morphology factors. Moreover, further research is preferred to examine the capacity of normalized ${\overline{\tau }}_E$ describing the ventilation capacity in realistic scenarios, considering complex building layouts and meteorological conditions. In addition, given the practical emission rate, the threshold values for normalized ${\overline{\tau }}_E$ could be calculated according to safe pollutant exposure levels.

4. Conclusions

This study adapted the mean age of air, a widely used nondimensional index in the field of indoor ventilation, to evaluate the impacts of building layouts on city ventilation conditions. Since a built environment is an open system, the adapted index was termed the effective mean age of air (${\overline{\tau }}_E$) for distinction. Four generic parameters that describe urban morphologies were considered in the CFD simulations for parametric study, with each group of cases focusing on one parameter: building height, building density, height variation, or the frontal area variation of adjacent buildings. Simulations were performed using the RANS method under an isothermal condition. The main conclusions of this study are listed below:

(1)

The distribution of poorly ventilated areas with a high ${\overline{\tau }}_E$ at the breathing level (z = 1.7 m) exhibits three primary patterns: within the recirculation zones behind buildings, in the downstream section of the main road, or in the recirculation zones next to the lateral facades of buildings. The relative strength between the main road flow and outward crossroads flow is essential to the pollutant transport direction and locations of accumulation sites at the breathing level.

(2)

Even if the average ${\overline{\tau }}_E$ of the focused area is similar between cases, the locations of high-${\overline{\tau }}_E$ regions and extreme values can vary significantly. The distribution features of the ${\overline{\tau }}_E$ data set efficiently reflect this information. Across all cases, the median of the average ${\overline{\tau }}_E$ is 32.75 s, and 62.18 s for the median of the 95th percentiles. These two values can be seen as thresholds distinguishing low, medium, and high ${\overline{\tau }}_E$ levels when the typical height of the urban region is 30 m.

(3)

Discussion also highlights that variations in building layouts do not always enhance ventilation through their disturbances to the flow. Conversely, abrupt discrepancies of adjacent buildings can lead to the presence of areas with an extremely high ${\overline{\tau }}_E$.

(4)

In general, convective transport is the primary contributor to the total purging for the canopy zone, while turbulent transport predominates for the pedestrian zone. Variations in the geometries of adjacent buildings slightly increase the turbulent transport rate, while changes in the overall building height and building density significantly alter the ratio between the two transport processes.

(5)

While decreasing the overall flow rate penetrating the built area, the presence of buildings has dual effects when focusing on local ventilation capacities. With a generic urban model with uniform 30 m-high buildings and equal spacings, at the breathing level, the main road is more polluted as expected, while the helical flows between buildings provide even better ventilation potential compared to the undisturbed parallel flow over an open space.

The findings and methodologies presented in this study provide diverse perspectives for further exploration and discussion in the urban planning process. According to the definition of Equation (1), the value of ${\overline{\tau }}_E$ is independent of the hypothetical emission rate, enabling the calculation of concentration at a specific location once actual emission data are provided. Furthermore, the homogeneous emission method represents a worst-case scenario in which pollutants are continuously released throughout the entire target region. Consequently, the distribution of ${\overline{\tau }}_E$ also reflects the local tolerance of the emission level during severe pollution events.