1 Introduction

Continuous increases in the population and limited land resources have caused rapid urbanisation all around the globe. According to Satterthwaite (2020), the urban population has increased to 56.2% of the global population in 2020 from 34% in 1960, with the urban count, as of 2020, of about 4.4 billion. This massive migration towards the cities has led to both vertical and horizontal expansion of the buildings, both in isolation as well as in clusters, to accommodate the ever-increasing population density. The interaction of urban buildings with the atmospheric flow significantly alters local conditions for pedestrian wind comfort (Tsang et al. 2012; Xu et al. 2017), surface temperature (Yang and Li 2015; Manoli et al. 2019; Li et al. 2020), and pollutant dispersion (Fuka et al. 2018; Marucci and Carpentieri 2020). As such, it clearly becomes important to study and understand the flow around tall buildings in urban environments. Numerical and experimental studies to understand such flow behaviour are frequently conducted on bluff cuboids, intended to offer simplified models of the actual buildings.

Extensive studies on single bluff body aerodynamics have been conducted (Castro and Robins 1977; Roshko 1993; Choi et al. 2008; Lim et al. 2009; Venugopal et al. 2011; Wang et al. 2019), focusing on the effects of various governing parameters such as building cross section (Sakamoto and Oiwake 1984; Moreau and Doolan 2013; Rostamy et al. 2013), aspect ratio (Agui and Andreopoulos 1992; Wang and Zhou 2009; Moreau and Doolan 2013), and boundary layer to building height ratio (Rostamy et al. 2013; Behera and Saha 2019). Castro and Robins (1977) studied the wake around a surface-mounted cube in a uniform shear layer and reported that the addition of upstream turbulence significantly reduced the size of the recirculation region behind the cube. They also observed that the velocity deficit decay in the wake showed a strong dependence on the turbulent intensities in the upstream flow. Martinuzzi and Tropea (1993) studied the effect of the lateral aspect ratio on the flow around prismatic objects in a fully developed turbulent channel flow. They reported a 2D flow field behind the objects with aspect ratio \((AR) > 6\), where, AR was defined as the ratio of cross-stream width to the height of the obstacle. They hypothesised that with an increase in AR, the two ’legs’ of the horseshoe vortex generated at the two corners are far apart and have little effect at the wake centre. Rastan et al. (2021) performed numerical simulations to study the effect of cross-sectional aspect ratio (CR, defined as the ratio of streamwise length to the cross-stream width of the model) on the wake dynamics. They observed reattachment of the flow separating from the leading edges for \(CR > 2\). El Hassan et al. (2015) conducted time-resolved particle image velocimetry (PIV) to study the effect of the turbulent boundary layer thickness on the wake dynamics of a square cylinder of \(AR = 4\). They observed that the horseshoe vortices significantly affect the wake flow, particularly when the boundary layer was tripped into turbulent flow. Studies by Castro and Robins (1977); Wang et al. (2006); Chen et al. (2022); Porteous et al. (2019); Behera and Saha (2021) have shown that the incoming boundary layer conditions have a profound effect on the wake behind wall-mounted cylinders.

Wake studies of 2D arrangements of multiple bluff bodies have also gained attention in the scientific community (Agrawal et al. 2006; Burattini and Agrawal 2013; Han et al. 2014; Zhang et al. 2019). PIV studies on three side-by-side square cylinders (More et al. 2020), circular cylinders (Sooraj et al. 2019), and six-in-line square cylinders (Sewatkar et al. 2012) have reported different flow regimes in the wake region depending on the distance between the cylinders. Sahu et al. (2022) studied the effect of side-by-side cylinders and a third cylinder placed upstream and then downwind on the boundary layer flow. They reported a decrease in turbulent intensity of 20% and turbulent kinetic energy by 15% when a third cylinder was kept upstream on the centreline of the two side-by-side cylinders. In the numerical studies conducted by Tang et al. (2019) on a square array of 10\(\times \)10 circular cylinders, a complete detachment of the recirculating wake from the array was observed within a given range of packing fraction and Reynolds number. Additionally, there are some recent investigations on the flow structures in the wake of a circular patch of circular cylinders (Chang and Constantinescu 2015; Taddei et al. 2016; Chang et al. 2017; Nicolai et al. 2020; Wangsawijaya et al. 2022). Chang and Constantinescu (2015) conducted numerical studies to understand the effect of solid volume fraction (SVF) on the flow characteristics in the wake of a circular array of circular cylinders. They observed that the flow is essentially 3D in most of the region for SVF > 0.1. They also noted that 3D features only occur downwind for SVF < 0.025. Similar studies were conducted by Chang et al. (2017) in an open channel to mimic the aquatic vegetation in a river channel. They observed the formation of a necklace vortex around the upstream cylinder for SVF > 0.2, which resulted in an increase in the magnitude of the mean and fluctuating bed friction velocity. They also found the drag forces acting on individual cylinders in the array to be strongly affected by the SVF. Nicolai et al. (2020) conducted PIV measurements in the lateral as well as vertical plane in the wake of patches of circular cylinders with different array densities. In their experimental investigation, they found a larger wake shed by the porous arrays when compared to the solid cylinder case. They also observed that the lateral wake behaved similarly to the classical planar wake, implying that the characteristics scales are the customary maximum velocity deficit and the wake half-width. Taddei et al. (2016) studied the relation between the drag coefficient and the wake of cylindrical canopy patches. They noted an increase in the drag coefficient of porous patches with increasing the SVF. From the experimental investigation of Wangsawijaya et al. (2022), it was found that while the horizontal extent of the wake decreases with an increase in SVF, the latter results in an increase in the vertical extent of the wake.

Although extensive studies have been conducted to understand the wake behind multiple bluff bodies, these have been mostly limited to 2D flows - i.e. by considering an infinite length of the cylinder in one direction (either spanwise or vertical). Additionally, finite wall-mounted bluff bodies have been considered for the study of the vertical wakes only (development of boundary layer behind bluff bodies). There have been some recent advancements in understanding the wake of porous obstacles. However, these have been limited to circular patches of circular cylinders, essentially mimicking the flow through aquatic or terrestrial vegetation. The present paper aims to understand the effect of different governing parameters on the horizontal wake of finite building clusters (i.e. square section obstacle arrays) which, to date, remain largely unexplored. The specific objectives are first to understand the magnitude and spatial scale of the effects of a cluster of tall buildings; second, to identify the main parameters that govern the extent and character of the near- and far- fields within the wake; and third, to assess what can be said generically of wakes and what remains site-specific. Regular square arrays of square cylinders are considered for this study, which resemble actual building shapes in urban areas. These are compared to the wake flow behind an isolated building of the same shape, which acts as a benchmark. The effects of building array size (N), building aspect ratio (AR), and building-to-building spacing (\(W_S\)) on the wake region are discussed in detail.

Fig. 1
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Schematic of the relative arrangements of Irwin spires, roughness elements and building models inside the wind tunnel: a schematic, and b actual setup

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Fig. 2
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a Top and b side view schematic of the cluster arrangements and building measurements

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Fig. 3
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a Reynolds shear stress distribution, and b comparison of mean velocity profile to the logarithmic law

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2 Methodology and Experimental Details

2.1 Experimental Facility

The present experiments were conducted in the ‘A’ wind tunnel of the EnFlo laboratory at the University of Surrey. This tunnel is of open circuit design having a working section dimension of 5 m \(\times \) 0.9m \(\times \) 0.6m (length \(\times \) width \(\times \) height) and can achieve a maximum flow speed of 25 m/s. A set of 7 Irwin spires of height 250 mm and width of 35 mm was placed at the inlet of the working section to generate an artificially thickened boundary layer. At the base, the length and width of the spire are 36 mm and 34 mm, respectively. The lateral spacing (centre-to-centre) between spires is 130 mm. The spires are followed by a 50% staggered array of roughness elements placed on the tunnel floor. Each roughness element is 2 mm in length and height and 8 mm in width. They are placed 16 mm apart along the spanwise direction and 24 mm apart in the streamwise direction (Fig. 1a inset). The cluster models are placed on a circular turntable of diameter 0.3m, 2980 mm downwind of the spires. The relative arrangements of the Irwin spires, the roughness elements, and the models on the turntable are shown in Fig. 1.

Table 1 Different cases of building clusters considered in the present study
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2.2 Test Cases

The building models for the present experiments are wooden cylinders of a square cross section of width (\(W_B\)) 10 mm and height (\(H_B\)) of 60 mm, providing AR (\(=H_B/W_B\)) of 6. To investigate the effect of AR, other clusters of building height 80 mm and 40 mm (AR = 8 and AR = 4, respectively) were also employed. Three different sets of cases are considered to study the effect of (i) cluster size (N), (ii) spacing between the buildings (\(W_S\)), and (iii) the building aspect ratio. To study the effect of the cluster size in isolation, arrays of size 3\(\times \)3, 4\(\times \)4, and 5\(\times \)5 are considered, with spacing between buildings equal to the building width (\(W_S = W_B\)), and \(AR=6\). The effect of aspect ratio is studied for a 5\(\times \)5 cluster having \(W_S = W_B\). Finally, the effect of building spacing is studied by varying \(W_S/W_B\) = 0.5, 1, 2, and 4 for 5\(\times \)5 cluster size, with \(AR = 8\). The details of each case are presented in Table 1. For comparison purposes, measurements were also taken for a building in isolation with the same cross section. Figure 2 shows the schematic of the arrangement of the building clusters and the coordinate system employed for the present experiment. A Cartesian right-hand coordinate system is employed, with x, y, z being the streamwise, spanwise and vertical directions, respectively. The origin is taken just downwind of the cluster in x, at the centre of the cluster in y and at the ground in z. Measurements were conducted in the wake flow behind the array in the (x, y) plane at the building mid-height (z = 0.5\(H_B\)), with boundary layer profiles taken in the (x, z) plane corresponding to the array centre line (y = 0).

2.3 Flow Measurements

Velocity measurements were acquired by a two-component laser Doppler anemometry (LDA) system (DantecDynamics), the lasers having a wavelength of nominally 532nm and 561nm. They were frequency-shifted using a 400 MHz Bragg-cell and are focused using a probe of diameter 27 mm at a focal length of 160 mm. The resulting measuring volume has a diameter of approximately 0.049mm and is 1.051mm long. The sample duration was 15 s for measurements at all points considered in the present work. The minimum sampling frequency was set to 100 Hz, controlled by the seeding system. 2D velocity data (UV) were recorded at each measuring location, both in lateral and vertical planes. A sugar particle aerosol of mean diameter 1 \({\mu }\)m was used to seed the flow for LDA. The tunnel working temperature was maintained at 20 \(^\circ \)C± 0.5 \(^\circ \)C.

A Pitot-static tube of a diameter of 2 mm was employed to record the free-stream flow speed in the tunnel. This was mounted 0.5m above the tunnel floor and at \(y=0.2\)m offset from the centreline to ensure that the effect of the walls and spires was negligible. The reference velocity \(U_{ref}\), as measured by the static-Pitot tube, was nominally 10 m/s. The boundary layer thickness (\(\delta \)) at x = 2980 mm from the spires was observed to be 223 mm, following the methodology given by Irwin (1981). The Reynolds number based on \(\delta \) and \(U_{ref}\) was 1.39 x \(10^5\). The friction velocity (\(u^*\)) is calculated from the Reynolds shear stress plateau so that \(u^*=\sqrt{\tau _0\rho ^{-1}}\approx \sqrt{-\overline{u'w'}}\) Cheng and Castro (2002). Here, \(\tau _0\) is the wall shear stress and \(\rho \) is the density of air. The distribution of Reynolds shear stress with height is shown in Fig. 3a, together with its plateau value (dash-dotted line). The roughness length (\(z_0\)) is then obtained by fitting the mean velocity profiles in logarithmic form for fully-rough conditions:

$$\begin{aligned} {\frac{U}{u^*}=\frac{1}{\kappa } ln \frac{z}{z_o}}, \end{aligned}$$
(1)

where \(\kappa =0.39\) is the von Kármán constant (Marusic et al. 2013). In the present work, \(u^*/U_{ref}=0.044\) and \(z_0/\delta =0.0008\), respectively, as shown in Fig. 3. The displacement length, d, is negligible and assumed to zero for the present experimental condition. The standard error in measuring the mean streamwise velocity was within \(\pm 1.5\%.\)

Fig. 4
figure 4

Wake flow behind a single building a non-dimensional vertical profiles of \(U/U_\delta \), b lateral profiles of \(U/U_0\), c wake growth, and d non-dimensional lateral profiles of \(V/U_0\)

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3 Results

3.1 Wake Flow Behind an Isolated Building–Single Building Case

Figure 4a shows the vertical variation of the non-dimensional mean streamwise velocity, \(U/U_\delta \), (where \(U_\delta \) is the local free-stream velocity outside the boundary layer at \(y = 0\)) at different streamwise locations in the wake developed downwind of a single building. The boundary layer profile in the absence of any building is also plotted for comparison and labelled ‘empty tunnel’. The empty tunnel boundary layer profile does not appear to significantly vary with the distance when measured \(x = 2500\)mm downstream of the spires, hence the empty tunnel boundary layer profile was captured at \(x = 3100\)mm from the spires. A negative mean velocity is observed immediately downwind of the building at \(x=W_B\) extending vertically to \(z=H_B\). As is well established, this flow reversal region is formed by the separation of the flow from the roof and sides of the building and the interaction of the resulting shear layers; this leads to lower pressure on the leeward side of the building. At \(z=H_B\), there is some ‘overshoot’ in these profiles due to acceleration over the building. The profiles at \(x=2W_B\), and further downwind, show momentum deficit (\(0< U/U_\delta < 1\)) behind the building typical of the wake flow, downwind of the recirculating region. The velocity deficit must be zero at the wall and at some height where the profile blends into the undisturbed boundary layer profile, resulting in a maximum deficit somewhere around \(z = 0.5H_B\). Measurements and theory reported in Counihan et al. (1974) show the same behaviour in the wake of 2D surface-mounted obstacles. In our case, the wake zone persists to about \(x = 20W_B\) and the velocity fully recovers (within experimental uncertainty) to that of the undisturbed state by \(x = 50W_B\).

The spanwise variation of the streamwise velocity is shown in Fig. 4b. The velocity data is taken at the mid-height of the building (\(z = 0.5H_B\)), to minimize the effect of any local flow features near the ground, such as the horseshoe vortex, as well as the shear layer developing at roof level. The lateral velocity variation in the absence of any building is also shown for comparison. As expected, negative velocities are observed behind the building at \(x = W_B\), within the recirculation region. The velocity deficit decreases downwind as the wake width increases, due to the momentum exchange in the lateral direction. Immediately downwind of the building (at \(x = 1W_B\) and \(x=2W_B\)) the velocity at the edge of the wake is observed to be greater than the local free-stream velocity (\(U_0\), defined as the local freestream velocity far away from the wake in y direction at \(z = 0.5H_B\) ); this is due to the local blockage of the building and the associated flow acceleration in the channels between the buildings. Far downwind (\(x = 50W_B\)), the wake decays completely, and the flow regains its undisturbed state (i.e., the upwind boundary layer flow as in Robins et al. (2018) and Oke et al. (2017).

Apart from the region of local acceleration above the building, the vertical extent of the wake remains at or below \(z = H_B\). In contrast, the wake spreads laterally, as characterised by the wake width (\({\sigma _y}\)). This is defined as the distance from the wake centreline to the lateral location where the deficit has recovered to 50% of the local maximum value (Liu et al. 2002; Khan et al. 2018). Figure 4c shows the growth of the wake, \({\sigma _y}\), in the spanwise direction behind a single building, as a function of x; for clarity, \({\sigma _y}\) for positive and negative y are shown separately. Both \({\sigma _y}\) and x are normalised by the building width, \(W_B\). The lateral wake growth is expected to be roughly linear in x, with \({\sigma _y}\) approximately the same on both sides for \(x \le 8W_B\). Herein, a non-symmetrical growth is observed. This asymmetry could be due to predominant vortex shedding in one direction or, more likely, due to a a slight misalignment of the building to the flow direction. The wake spread rate, \(\frac{d\sigma _y}{dx}\), was found to be 0.14 and 0.18 in the +ve y and -ve y, respectively. We also calculated the spread in the vertical wake (\({\sigma _z}\)) by first obtaining the velocity deficit with respect to the velocity at the same height for the undisturbed case. \({\sigma _z}\) was then taken as the height at which the deficit became half of the maximum deficit. A decay of the vertical wake scale (\({\sigma _z}\)) is observed in Fig. 4c, as might be expected given the geometry and its definition. However, to note that the depth of the affected flow extends up to, and above, the building height. Far downwind, the profiles merge into that of the boundary layer in the undisturbed state.

To complement the information on the streamwise velocity and to visualise the flow behaviour in the spanwise direction, the variation of spanwise velocity (V) in the lateral direction (y) is reported in Fig. 4d. In the near region behind the building, V shows an anti-symmetric profile about y = 0, which is expected, as the wake decay implies inflow towards the wake’s centre. The lateral velocity deficit decreases in the downwind distance more rapidly than that of the mean velocity and it is small beyond \(x \approx 8W_B\). Interestingly, the velocity profiles at downwind of \(x = 8W_B\) indicate the flow moving away from the building. This behavior can perhaps be attributed to the vortex shedding phenomenon, although the magnitude of the spanwise velocity is relatively small (less than 2% of \(U_0\)), hence it is affected by higher experimental uncertainty as the signal-to-noise ratio deteriorates.

Fig. 5
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Lateral profiles of non-dimensional streamwise velocity for Case N5 in the a near-wake regime, b transition-wake regime, and c far-wake regime

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Fig. 6
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Lateral profiles of non-dimensional spanwise velocity for Case N5 in the a near-wake regime, b transition-wake regime, and c far-wake regime

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Fig. 7
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Lateral profiles of non-dimensional streamwise velocity for Case N4, \(W_S=W_B\) (a, b) and Case N3, \(W_S=W_B\) (c, d) in near-wake regime (a, c), transition, and far-wake regime (b, d)

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Table 2 Summary of wake regimes and flow region of interest
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3.2 Wake Development Behind Clusters of Building

Unlike the wake flow behind a single building, which is reasonably well understood – at least for cuboids (Castro and Robins 1977; Robins et al. 2018), the study of the more complex wake developing behind a cluster of buildings is more limited. The effect of the individual buildings is expected to dominate the physics just immediately downwind of the array, while the individual wakes subsequently merge to form a single global wake, which can be hypothesised to be better characterised by the size of the array, as further discussed in this section. Figure 5 shows the spanwise variation of streamwise velocity behind a cluster of size 5\(\times \)5 (Case N5). Three different wake profile types are observed and are plotted separately in Fig. 5. Close observation shows the formation of individual wakes behind each building in the near field in Fig. 5a). Due to channelling effects, the flow gains momentum in the space between the buildings. These individual wakes and ’jets’ interact with each other further downwind (Fig. 5b), eventually merging into a single global wake structure (Fig. 5c).

In more detail, the velocity profile at x = 0.1\(W_A\) (Fig. 5a) shows the velocity deficit behind each individual building. Acceleration in the spacing between the buildings is also observed due to the channelling effect. An interesting phenomenon to note is that the velocity deficit has minima just behind the buildings which are at the edge of the array, while it increases towards the centre. This phenomenon indicates that wake interaction between buildings is more prominent towards the centre. Subsequently, there is a higher gain in the velocity in the channels towards the centre of the array due to momentum conservation. As the individual building wake moves downwind, it begins to interact with the adjacent wakes. The wake profile at x = 0.22\(W_A\) (Fig. 5a) shows that the velocity deficit behind the buildings is decreasing at the expense of the velocity excess in the channels. This clearly indicates momentum exchange in the lateral direction. Due to this exchange, the individual wakes begin to merge, forming a single global wake far downwind. The merging phenomenon is shown to take place between x = 0.44\(W_A\) and 1.33\(W_A\) in Fig. 5b. The global wake is not yet fully developed at these locations which can be categorized as a ‘transition zone’. Here the term ‘transition’ is used to signify flow states between those of individual building wakes and those of a fully-developed global wake. The latter is observed at x = 2.22\(W_A\) and 5.56\(W_A\) (Fig. 5c).

The lateral variation of the spanwise velocity, V, also clearly identifies the three regimes as is shown in Fig. 6. In the near-wake regime (Fig. 6a), we observe alternate maxima and minima due to the processes associated with the decay of the individual wakes (see Fig. 4d) superimposed on the ’global’ variation across the whole array. Merging of wakes is observed in the transition regime (Fig. 6b), which then develops into a global-wake regime (Fig. 6c). In this regime, the clusters behave similarly to a single building, provided that their wake is appropriately scaled.

Fig. 8
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Lateral profiles of non-dimensional streamwise velocity for clusters with different building aspect ratio - Case A8 (a, b) and Case A4 (c, d) in the near-wake regime (a, c), transition-wake, and far-wake regime (b, d)

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Fig. 9
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Lateral profiles of non-dimensional streamwise velocity in the near- (a, c, and e); transition- and far-wake regions (b, d, and f) for 5\(\times \)5 array at different building spacings: Case W05 (a, b), Case W2 (c, d), and Case W4 (e, f)

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Fig. 10
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Vertical profiles of non-dimensional streamwise velocity downwind the building clusters: (a) Case N5, (b) Case N3, (c) Case W05, (d) Case W2, and (e) Case N4

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3.2.1 Effect of Cluster Size

To investigate the effect of the array size on the spatial extent of the near-, transition-, and far-wake regions, experiments were also conducted for the 4\(\times \)4 (Case N4) and 3\(\times \)3 (Case N3) array sizes, keeping other quantities constant (Table 1). Streamwise velocities are shown in Fig. 7. Again, individual wakes form behind each building, at x = 0.14\(W_A\) and 0.29\(W_A\) for Case N4 (Fig. 7a), and at 0.2\(W_A\) and 0.4\(W_A\) for Case N3 (Fig. 7c). The flow transitions to the far-wake regime at around x = 1.71\(W_A\) for Case N4 (Fig. 7b) and x = 1.6\(W_A\) for Case N3 (Fig. 7d). It should be noted that for Case N4, the array centre is not behind a building, but in the channel between two buildings. Hence, we observe gain in velocity at y = 0 for evenly-sized building clusters. The array centreline velocity for Case N4 is expected to behave differently from the other cases in the near-wake regime as the centre of the array lies in the channel between the buildings, as discussed later in Sect. 3.3.1 (see, for example, Fig. 11a). However, self-similar wake characteristics are observed in the far-wake region irrespective of the cluster size, allowing for this comparison between different array sizes.

3.2.2 Effect of Building Aspect Ratio

The lateral velocity profiles for 5\(\times \)5 building clusters with different AR (case A8 and A4) are shown in Fig. 8. The velocity profiles show a similar trend at different AR. Comparing Figs. 8 and 5 (Case N5) shows that at \(x = 0.1W_A\) and 0.22\(W_A\), maxima and minima in the velocity profiles are observed due to the individual wakes forming behind each building. At \(x = 0.44W_A, 0.9W_A\), and \(1.33W_A\), (Fig. 5b, 8b, and d) the wakes are observed to merge. It is interesting to note that the location at which the transition region begins appears to be invariant of the aspect ratio of the individual buildings. A single global wake is then observed for \(x \ge 2.22W_A\). Although we did not observe any significant effect of buildings’ height on the wake regimes, it is expected that with a further increase in their relative height compared to the boundary layer depth, the effect of the rooftop shear layer and the ground level vortices will be visible further downwind; this may reduce the wake recovery rate, extending the different wake regimes.

3.2.3 Effect of Building Spacing

To observe the effect of building packing density on the wake flow, the 5\(\times \)5 building array was investigated with different spacings between buildings (so that \(W_s\)/\(W_B\) = 0.5, 2 and 4). The profiles of mean streamwise velocity are presented in Fig. 9. The near-wake flow is seen to be significantly influenced by the change in building spacing. Case W05 has the highest packing density with \(W_S\)=0.5\(W_B\). Individual wakes behind each building are observed for \(x = 0.14W_A\), and \(0.29W_A\) (Fig. 9a) but, due to the reduced spacing, the channelling effect is only prominent near the centre of the array. It is hypothesised that the displacement of flow in the vertical direction dominates in this case, although this cannot be fully supported based on current data. The experiments by Nicolai et al. (2020) have shown an increase in vertical bleeding of the flow for patches with high SVF. Further, Wangsawijaya et al. (2022) have found out that at high SVF, the flow gets trapped within the patch and escapes largely through vertical bleeding. The merging of the wakes in the transition regime is observed at \(x= 0.57W_A\) (Fig. 9b), which eventually develops into a global wake structure.

When the spacing between the buildings is sufficiently increased, the wake-jet pattern becomes more prominent, which is well captured by the wake profiles in Case W2 (Fig. 9c) and Case W4 (Fig. 9e). Compared to Case N5 (Fig. 5), the differences in the magnitude of the velocity deficit behind each building reduces for Case W2 (at x = 0.08\(W_A\)) and becomes almost negligible in Case W4 (at x = 0.04\(W_A\)); this indicates a significant reduction in the wake interaction just downwind of the clusters. It is interesting to note that negative velocities are observed, for each case, at the nearest location behind the array, highlighting that the strength of the recirculation region behind the clusters is governed by the geometry of the individual buildings. A detailed investigation, however, would be required to comment on the effect of \(W_S\) on the extent of the recirculation zone. To note that for Cases W2 and W4, we observed a flat velocity profile in the transition- wake regime (Fig. 9d and f, respectively); this is due to the larger cluster size. It is interesting that for all the cases considered herein, the extent of the three distinct zones can be quantified in terms of the cluster size, as defined in Table 2. It is, however, speculated that with an increase in \(W_A/\delta \), the lateral entrainment of freestream flow towards the wake centre will reduce when compared to the vertical entrainment, which may lead to an increase in the streamwise extent of the wake regions, particularly within the transition- and far-wake regimes.

The vertical structure of the mean velocities downwind of the arrays is shown in Fig. 10. The empty tunnel boundary layer profile is also shown for reference (labelled as ‘ET’). Measurements below \(z = 0.33H_B\) were not possible due to experimental limitations. Note that the centreline (\(y = 0\)) for all cases other than Case N4 (Fig. 10e) lies just downwind of a building. A recirculating flow region is observed behind the buildings, similar to the single building case in Sect. 3.1, and extending vertically to the building roof height. The velocity deficit recovers downwind, eventually returning to the undisturbed state (or nearly so). There are differences in details though. Close observation of the velocity profile at distance corresponding to the nearest location downwind of the array (for each case) shows two different flow patterns. For Case N5 and Case W2 (Fig. 10a and d), the velocity in the cavity region is almost constant across the building height. A slight increase in the velocity near the ground is instead observed for Case N3 and W05 (Fig. 10b and c), which is possibly due to the interaction of the horseshoe vortices and the arch vortices on the leeward side of the array. It can be hypothesised that the effect of these vortices is dominant near the array centre for small arrays. With an increase in the array size (i.e. with increasing N), the interaction of these vortices with the wake centre occurs further downwind. Their effects are , therefore, seen in the downwind location. These results further support the hypothesis that the size of the clusters has a strong influence on the location of wake merging and development of the global wake. The boundary layer grows distinctively in each wake regime. In the near-wake regime, the profile is near-uniform over much of the wake depth. In the transition-wake regime, there is little or no significant velocity change with streamwise locations. The boundary layer then progressively recovers to the undisturbed state in the far-wake regime. Unlike in the other cases, the centreline for Case N4 lies in the channel between the buildings (Fig. 10e; see also Fig. 7a and  b), within the jet-like flow, however, the mean velocity deficit remains positive when referred to the undisturbed profile. The recovery is more complex than in a single building wake, as there is first a region dominated by the jet flows in which the mean velocity decreases (see profiles at \(x =0.14\) to \(0.57W_A\)) before beginning to increase again. A transition region between these two behaviours can be seen in the profiles at \(x =1.14\) and \(1.71W_A\). The flow is accelerated in the channel between the buildings, as observed from the velocity profile at x = 0.14\(W_A\), and 0.29\(W_A\). The velocity reduces downwind up to x = 1.14\(W_A\), due to the exchange of the energy with the wakes formed behind the individual buildings. As the individual wakes coalesce forming a global wake, the velocity increases again until it reaches its fully-developed state.

Fig. 11
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Effect of geometrical parameters on the decay of the centreline velocity deficit, (a) array size, (b) aspect ratio, and (c) building spacing

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Fig. 12
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Velocity scaling laws in the (a) far-wake regime, and (b) near-wake regime

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3.3 Wake Scaling Behind Building Clusters

It has been established in Sect. 3.2 that the wake behind the cluster of buildings can be categorized by three different wake regimes (Table 2), each having distinct flow features. This section discusses the different parameters governing the flow characteristics in the near- and far-wake regimes.

3.3.1 Centreline Velocity Scaling

Figure 11 shows the influence of array geometry on the decay of the centreline velocity deficit (\(\varDelta U_c= U_0 - U_c\), where \(U_0\) is the local freestream velocity and \(U_c\) is the centreline velocity measured at \(y=0\) at a given streamwise location) downwind of the building clusters. With an increase in the downwind distance, the wake spreads laterally and therefore, the velocity deficit decreases. The velocity deficit is observed to depend on array size, aspect ratio, and the distance between buildings within the array. The effect of array size is shown in Fig. 11a. Note that for 4x4 array size (case N4), the centreline velocity is not directly behind a building, but in the channel between buildings. Therefore, it exhibits an opposite behaviour in the near-wake regime when compared with odd-sized arrays (Case N5 and N3). First, the steep decay/rise in \(\varDelta U_c\) for Case N5 & N3/Case N4 in the near wake is expected. The proximity in the array wake implies momentum transfer from the jets to the wakes. In the far-wake regime, the centreline velocity deficit, \(\varDelta U_c\), is larger for Case N5, and a decreases with decrease in the array size (case N4 and N3).

The effect of aspect ratio on the decay of \(\varDelta U_c\) on the centreline of a 5x5 array is shown in Fig. 11b. As before, a steep decay in \(\varDelta U_c\) is observed in the near-wake regime is observed, here, however, in a jet region. Interestingly, a slight increase in \(\varDelta U_c\) is observed in the transition regime, as the internal structure of the wake adjusts across the near- and far-wake regions. The velocity deficit in the far-wake regime increases with an increase in AR. As noted by Kimura et al. (2018), the recovery of the wake is significantly influenced by the generated longitudinal vortices. The horseshoe and rooftop vortices in an array with smaller AR have a greater influence on the array centre, thus facilitating a faster wake recovery.

The decay rate of the centreline velocity deficit is observed to be highly sensitive to the spacing between the buildings constituting the cluster (Fig. 11c). Interestingly, in building clusters with \(W_S\)=0.5\(W_B\), there is a significant increase in the deficit between x = 50 mm and x = 150 mm. A plausible explanation could be that the streamwise velocity reduces in the transition regime at the expense of an increase in vertical velocity. However, a detailed 3D flow field analysis would be required to confirm this phenomenon.

The recovery of the centreline velocity in the wake behind a square cylinder is well understood (Lyn et al. 1995; Rodi 1997; Saha et al. 2000). As the wake behind clusters shows a complex transition from multiple individual wakes (which scale with the dimension of the buildings as discussed in Sect.  3.2) to a single global wake (scaling with the dimension of the cluster as a whole), the relevant lengthscales are expected to be \(W_B\) and \(W_A\) in the near- and far-wake regions, respectively. This is clearly illustrated by Figs. 12a and b, which present the decay of velocity deficit with respect to the local freestream velocity (\(U_0\)) and average channel velocity (\(U_{loc}\)), respectively. \(U_{loc}\) is defined as the average of the velocities at the centreline of the channels between the buildings at a given streamwise location. The decay in the near-wake regime is calculated as the difference in the average velocity in the channels and the velocity at the centreline (\(y=0\)). It is observed that in the near-wake regime, the decay of the velocity deficit is governed by the array size (N), the space between the buildings (\(W_S\)), and the building width (\(W_B\)). The velocity deficit in the far-wake regime is characterised by the building aspect ratio (AR) and the frontal-area density (\(\lambda _f=\frac{NHW_B}{HW_A}\)).

Fig. 13
figure 13

Lateral profiles of non dimensional velocity deficit in the far-wake regime: a Case SB, b Case N5, c Case N4, d Case N3, e Case W05, and f Case W2

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Fig. 14
figure 14

Lateral profiles of dimensionless velocity in the near wake regime: a Case N5, b Case N4, c Case N3, d Case W05, e Case W2, and f Case W4

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Fig. 15
figure 15

Streamwise decay of normalised local velocity deficit

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3.3.2 Wake Profiles Scaling

The behaviour of streamwise velocity in the far- and near-wake regimes are represented in appropriate scalings in Figs. 13 and 14, respectively. It is observed that in the far-wake regime, the velocity deficit profile (normalised with the centreline velocity deficit, \(\varDelta U_c\)) at any streamwise location (\(x/W_A\)) achieves concurrence when the lateral distance y is normalised with the wake half-width, \(Y_{0.5}\). This is defined as the distance from the array centre in the spanwise direction at which the velocity deficit becomes half of the maximum velocity deficit. Note that \(\varDelta U_c\) and \(Y_{0.5}\) give good collapse in the far-wake regime (\(x/W_A > 1.5\)) for all the cases considered in the present experiment (Fig. 13). This indicates that the wake in the far-wake regime for clusters of building behaves similarly to that of a single isolated building (Fig. 13a).

The scaling parameters in the near-wake of building clusters are difficult to determine since the effect of the individual buildings and the channelling between them are prominent in this region. Therefore, instead of using a global velocity deficit, we propose a new dimensional scaling for the near-wake regime that is based on a local velocity deficit, \(\varDelta U_{loc}\). This is defined as:

$$\begin{aligned} {\hat{U}}= \left\{ \begin{array}{ll} \frac{U-{U_{cl}}}{U_{loc}-{U_{cl}}}, &{}\Vert y\Vert {\le } 0.5W_A,\\ \\ \frac{U-{U_{c}}}{U_0-{U_{c}}}, &{}\Vert y\Vert {>} 0.5W_A,\\ \end{array} \right. \end{aligned}$$
(2)

Within \(y = \pm 0.5W_A\), the local velocity deficit, \(\varDelta U_{loc}\), is defined as the difference between the average velocity in the downwind projection of the channels between the buildings (\(U_{loc}\)) and the average velocity in the wakes behind buildings (\(U_{cl}\)). The velocity profiles in the near-wake regime (\(x \le 0.5W_A\)) collapse when dimensionalised in this manner. The spanwise direction, here, is normalised with the array width, \(W_A\). As observed in Fig. 14, this new scaling achieves a good concurrence in the velocity across all cases. It can be established that in the near-wake regime, the individual building wakes are dominant; these eventually merge to form a single wake resembling that behind a single building of the same geometry.

The normalised variation of \(\varDelta U_{loc}\) with streamwise distance for the eight test cases is shown in Fig. 15. In each case, the local velocity deficit has been normalised by \(\varDelta U_c\) at the same streamwise location. This leads to a satisfactory collapse of the near-wake data. Also of note is that the velocity deficit decreases to zero in the near- wake regime and becomes constant in the transition- and the far-wake regime, where the individual jets and wakes have fully merged.

4 Conclusions

The present paper aimed to understand the characteristics and the scaling laws of the wake flow behind clusters of buildings of different array sizes, aspect ratios, and building spacing immersed in deep boundary layers. 9 different test cases are considered with cluster sizes from 1x1 to 5x5. Tests were conducted in a wind tunnel and streamwise and spanwise velocity components were acquired using 2D LDA.

Results have shown that –irrespective of the parameters investigated– the wake exhibits different flow behaviour in different regions behind the building clusters. In the near-wake region immediately behind the array (\(\textit{x} \le {0.45W_A}\)), separate wakes are observed to form behind each individual building. The flow between the buildings gains momentum due to a channelling effect. A transition- wake regime was observed between \(0.45W_A \le \textit{x} \le 1.5W_A\). In this transition-wake regime, the individual building wakes merged to develop into a single, larger wake. The merging phenomenon was found to start at the array edge and then move towards the centre of the array. Finally, the cluster wake developed into a single wake in what was termed the far-wake regime (\( \textit{x} \ge 1.5W_A\)); this was found to be similar to the wake behind a single isolated building of the same geometry. These three wake regimes were observed to be governed only by the cluster array width, \(W_A\).

It must be highlighted that the distinction between the three wake regimes is currently limited to clusters organised in a regular array with a wind angle of 0 \(^\circ \). However, asymmetrical wakes are expected for wind angles other than 45 \(^\circ \), and therefore, the flow features within the near-wake regime might substantially change. Furthermore, the spacing between the buildings is also expected to be an essential variable, with the flow physics likely to be governed by the frontal blockage; additional data is however needed to sufficiently support this hypothesis. Another limitation when generalising this work is the asymptotic behaviour for large and small building spacing, \((W_S)\). When the spacing between the buildings is either too small or too large, individual building wakes may not distinctly develop, affecting the dominant features of the near-wake region.

The decay of the centreline velocity deficit (\(\varDelta U_c\)) shows a dependence on array size, aspect ratio, and the spacing between the buildings. Different decay rates of \(\varDelta U_c\) were observed in the near-wake regime and the far-wake regime. In the near wake, the momentum is transferred from the channels between the buildings into the wake, resulting in a decrease in the velocity deficit behind the buildings and an increase in the velocity deficit in the channels. The flow dynamics in the transition wake regime showed complex behaviour, with an outward transfer of the momentum from the wake towards the freestream region. In the far wake (termed far-wake regime), the wake deficit is observed to increase with the array size (N) and the building aspect ratio (AR). Smaller array sizes facilitate the interaction of the vortices with the wake, resulting in faster recovery of the wake. The velocity scaling laws showed that in the far-wake regime, the decay of the centreline velocity with respect to the freestream velocity is governed by frontal blockage, building aspect ratio, and array width. On the other hand, in the near-wake regime, the decay of the centreline velocity with respect to the average channel velocity is governed by the array size, building width, and the spacing between the buildings. The local channel velocity is observed to be an important governing parameter in the near-wake regime. The variation of local channel velocity deficit showed good collapse for all the data considered in the present experiment when scaled with centreline velocity deficit (\(\varDelta U_c\)) in the near-wake regime.

The present results allowed an enhanced understanding of the development and characteristics of different wake regimes behind a cluster of tall buildings aligned in the wind direction. It does, however, offer scope for further studies, particularly to understand the effect of relative building height (\(H_B\)) and array size (\(W_A\)) with respect to the boundary layer thickness (\(\delta \)) on the extent of the different wakes.