# Spatial Characteristics of Roughness Sublayer Mean Flow and Turbulence Over a Realistic Urban Surface

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## Abstract

Single-point measurements from towers in cities cannot properly quantify the impact of all terms in the turbulent kinetic energy (TKE) budget and are often not representative of horizontally-averaged quantities over the entire urban domain. A series of large-eddy simulations (LES) is here performed to quantify the relevance of non-measurable terms, and to explore the spatial variability of the flow field over and within an urban geometry in the city of Basel, Switzerland. The domain has been chosen to be centered around a tower where single-point turbulence measurements at six heights are available. Buildings are represented through a discrete-forcing immersed boundary method and are based on detailed real geometries from a surveying dataset. The local model results at the tower location compare well against measurements under near-neutral stability conditions and for the two prevailing wind directions chosen for the analysis. This confirms that LES in conjunction with the immersed boundary condition is a valuable model to study turbulence and dispersion within a real urban roughness sublayer (RSL). The simulations confirm that mean velocity profiles in the RSL are characterized by an inflection point \(z_{\gamma }\) located above the average building height \(z_\mathrm{h}\). TKE in the RSL is primarily produced above \(z_{\gamma }\), and turbulence is transported down into the urban canopy layer. Pressure transport is found to be significant in the very-near-wall regions. Further, spatial variations of time-averaged variables and non-measurable dispersive terms are important in the RSL above a real urban surface and should therefore be considered in future urban canopy parametrization developments.

## Keywords

Large-eddy simulation Turbulence Turbulent kinetic energy budget Urban canopy Urban roughness sublayer## 1 Introduction

Accurate modelling of flow and turbulence in the urban roughness sublayer (RSL), the atmospheric layer from the ground to 2–5 times the average building height \(z_\mathrm{h}\), is essential to predicting weather, air quality, and the dispersion of gases in the urban environment. Within the RSL, flow and turbulence exhibit strong spatial variations in both the vertical and the horizontal directions, variations that are caused by the flow around the local configuration of roughness elements (buildings and trees). Hence, one-dimensional surface scaling relying on horizontal homogeneity such as the Monin–Obukhov similarity theory (MOST) is not applicable in the RSL (Rotach 1999; Roth 2000). MOST is strictly applicable only in the inertial sublayer (ISL), whose existence in urban environments is subject to debate (Jimenez 2004). Consequently three-dimensional approaches such as computational fluid dynamics (CFD) are required to properly describe flow, turbulence and vertical exchange in the RSL.

However, for many applications, building-resolving information is neither required nor are CFD approaches computationally feasible. In mesoscale weather forecasting and air pollution dispersion models, urban canopy parametrizations (UCP) are used to represent the effects of urban surfaces. UCPs rely usually on a horizontally-averaged approach, where the RSL is represented as a 1D column, often for simplified geometries such as infinite street canyons or cubical blocks of buildings. The vast majority of UCPs use MOST relationships to compute vertical fluxes of momentum and scalars such as heat, humidity and pollutants between the urban facets and the atmosphere, irrespective of the problems outlined above (Grimmond et al. 2010).

Proper techniques to reintroduce a 1D approach in a truly three-dimensional RSL should account for the inherently variable canopy morphology, and its hierarchical structure of scales (from the street or canyon scale to the regional scale) as discussed in Britter and Hanna (2003). For instance, in the horizontal averaging process of the Reynolds-averaged Navier–Stokes (RANS) equations, additional terms arise in the time-averaged momentum balance, called *dispersive fluxes* (Raupach and Shaw 1982), which physically represent spatial correlations between mean vertical flow around buildings and the time-averaged quantity exchanged. The very few modelling studies directly determining dispersive fluxes by means of CFD have shown that these terms can be highly relevant, in addition to Reynolds stress, to the overall momentum transfer in the RSL over rigid canopies (Coceal et al. 2006; Martilli and Santiago 2007).

From a fundamental perspective, efforts using experimental and numerical approaches have been devoted to studying RSL dynamics and scalings over simplified urban-like surfaces, mostly in the form of staggered/aligned cubical arrays (Cheng and Castro 2002; Xie and Castro 2006; Coceal et al. 2006; Cheng and Porté-Agel 2013, 2015; Anderson et al. 2015). The few characteristic length scales that characterize roughness elements in such arrays provide a setting that simplifies simulation, analysis and the development of theory. The approach is justified on the grounds that one should first understand flow over rough surfaces in its simplest form, before introducing complexities such as variable roughness height or shapes, which would result in a broader spectrum of scales and dynamics. However, flow over cubes might be difficult to compare with flow over real urban canopies, where the additional set of length scales, connected to the intrinsic heterogeneity of the surface, might completely modify the dynamics of the system. For instance, boundary-layer flow over surface-mounted cubes with variable element heights, Cheng and Castro (2002) report a thinner ISL when compared with uniform height settings, suggesting an ISL region might not even exist in certain realistic urban canopies. Recent simulations of flow over cubes (Yang et al. 2016) have shown that at high Reynolds numbers, the mean velocity profiles exhibit exponential and logarithmic layers, even for cases with a considerable range of varying cube heights. Further, the effects of building representation and clustering in flows over realistic urban canopies also influence the dynamics of the system (Bou-Zeid et al. 2009).

In the past few decades experimentalists have devoted significant efforts to measuring the relevant processes that drive mean flow and turbulence in the RSL over real cities (Grimmond and Oke 1999; Eliasson et al. 2006; Christen et al. 2007; Ramamurthy et al. 2007; Christen et al. 2009; Peng and Sun 2014; Wang et al. 2014; Ramamurthy and Pardyjak 2015). However, such field studies are limited to measurements at a few points and cannot capture the full three-dimensional flow field in its heterogeneous state. The lack of homogeneity in the statistical properties of the flow within the RSL raise questions on the use of point measurements as a surrogate of horizontally-averaged quantities, as proposed by Rotach (1993a, b) and Christen et al. (2009). The strong spatial variability of the flow represents in fact the main challenge preventing the development of a comprehensive physically-based theory for the vertical structure of the RSL, such as the classic similarity approach (Monin and Obukhov 1954) for the idealized surface layer.

The increased availability of high resolution digital datasets on urban morphology (e.g. high resolution lidar scans, vectorial models based on surveyed data, etc.) encourages the use of real topographies in CFD studies (see for instance Kanda et al. 2013). Further, advances in computational power now allows the representation of the three-dimensional processes of interest at the neighbourhood scale (\(\mathcal {O}(10^2-10^3) \ \mathrm {m}\)). This is at least allowing constraints to be relaxed with regard to the feasibility and cost of numerical simulations over real urban morphologies.

Output from numerical models, such as large-eddy simulation (LES), can be used to understand the physics of the flow and quantify the most relevant terms and processes that occur in a realistic urban RSL. This is the goal of the current study. Here LES is used to resolve the airflow over and within a detailed urban geometry to, (1) spatially characterize mean flow and turbulence in the RSL, (2) to determine the role of non-measurable terms such as dispersive momentum fluxes, wake production, dispersive transport, pressure transport, dissipation of turbulent kinetic energy (TKE), and (3) to determine how representative are single-point measurements, when used as a surrogate for horizontally-averaged quantities over the entire urban domain. Such information can then be used to guide and validate current upscalings for one-dimensional UCPs.

Throughout the study the Einstein notation is alternated with the vector notation, based on convenience, with *x*, *y*, *z* denoting the streamwise, spanwise and vertical coordinates. The boundary-layer height is denoted as \(\delta \) whereas a given height in the domain is \(z_\mathrm{label}\), where the subscript “label” refer to various specific heights. Further, \(\widetilde{(\cdot )}\) is used to denote a spatially filtered variable (the spatial filtering that is implicitly understood in LES), \(\overline{(\cdot )}\) is time-averaging or ensemble averaging (depending on the context), \(\langle \cdot \rangle \) is horizontal (*x*, *y*) averaging, time fluctuations are written as \((\cdot )^{\prime }\) (therefore \(\overline{(\cdot )^{\prime }} = 0\)) and departures of time-averaged terms with respect to their horizontal mean are denoted as \(\overline{(\cdot )}^{\prime \prime }\) (therefore \(\langle \overline{(\cdot )}^{\prime \prime } \rangle = 0\)); \((\cdot )^*\) denotes a normalized variable.

## 2 Materials and Methods

The LES approach is based on the assumption that the energy containing scales of the flow are explicitly resolved. These large-scale motions are the main contributors to the transport of momentum, but due to their strong dependence on boundary conditions and to their intrinsic anisotropy, their effects are difficult to parametrize, typically leading to complex RANS closure models. LES aims instead at providing an adequate model for the “small scales” of the flow, ideally belonging to the inertial subrange of turbulence (Meneveau and Katz 2000), which allows simple parametrizations to be very effective, and it is implicitly assumed that the large-scale motions are properly resolved by the chosen numerical scheme.

### 2.1 Numerical Algorithm

The LES algorithm has been previously used to study land-atmosphere interaction processes (Albertson and Parlange 1999a, b) and to develop and test linear and non-linear LES SGS models (Meneveau et al. 1996; Porté-Agel et al. 2000; Porté-Agel 2004; Bou-Zeid et al. 2005; Lu and Porte-Agel 2010, 2013).

Equations are solved in strong form on a regular domain \(\varOmega \), a pseudo-spectral collocation approach (Orszag 1969, 1970) based on truncated Fourier expansions is used in the *x*, *y* coordinate directions, whereas a second-order accurate centered finite differences scheme is adopted in the vertical direction, requiring a staggered grid approach for the \(\tilde{u},\tilde{v},\tilde{p}\) state variables (these are stored at \((j+1/2)\mathrm{d}z\), with \(j=1,nz\)). Time integration is performed adopting a fully explicit second-order accurate Adams-Bashforth scheme and a fractional step method (Chorin 1968; Kim and Moin 1985) is adopted to compute the pressure field, which is based on an operator-splitting technique. In addition, non-linear terms are deliased via the 3 / 2 rule (Canuto et al. 2006), to avoid the piling up of energy in the high wavenumber range (Kravchenko and Moin 1997). The computational boundary is partitioned as \(\partial \varOmega = \varGamma _{\mathrm {b}} \cup \varGamma _{\mathrm {top}} \cup \varGamma _\mathrm{lateral}\), where \(\varGamma _{\mathrm {top}}\) and \(\varGamma _\mathrm{lateral}\) denote the top and lateral boundaries respectively. A free-lid boundary condition applies at \(\varGamma _{\mathrm {top}}\) and a parametrized boundary condition is prescribed at \(\varGamma _{\mathrm {b}}\) (see in Eq. 1). Periodic boundary conditions apply at \(\varGamma _\mathrm{lateral}\) due to the Fourier spatial representation.

#### 2.1.1 Subgrid-Scale Closure Model

The proposed study considers two LES closure models to evaluate \(\tau _{ij}^{\mathrm{SGS}}\): the classical static Smagorinsky model (Smagorinsky 1963) in conjunction with a wall damping function (SMAG), similar to that adopted in Mason and Thomson (1992), and the scale-dependent model with Lagrangian averaging of the coefficient (LASD), developed in Bou-Zeid et al. (2005).

The SMAG model prescribes a constant coefficient, whose value is usually that derived from the theory of homogeneous turbulence \((c_{\mathrm{s},\varDelta }=0.16\), for the sharp spectral cut-off filter). However, in applications involving high Reynolds number boundary-layer flows, such as that proposed herein, the model is known to be over-dissipative in the near wall regions, where \(c_{\mathrm{s},\varDelta }\) should approach zero. To cope with this we introduce an empirical wall damping function (Mason and Thomson 1992), which has the drawback of requiring an ad hoc calibration for each specific flow case, but partially ameliorates the dissipative properties of the SMAG model.

The LASD model overcomes the necessity of ad hoc specification of the damping function by exploiting the smallest resolved scales to compute the model coefficient at runtime. It represents an evolution of the original dynamic model, based on the Germano identity (Germano et al. 1991) and its modifications (Lilly 1992). LASD relaxes the scale invariance assumption of the model coefficient, which is a desirable property in the near wall regions, where the grid size approaches the limits of the inertial subrange (Meneveau and Katz 2000). The Lagrangian averaging of the model coefficient makes the model well suited for applications involving complex geometries, since it preserves local variability while satisfying Galileian invariance, and overcomes the requirement of homogeneous directions (Bou-Zeid et al. 2005). Additionally, the energy cascade process is more apparent along fluid pathlines (Meneveau and Lund 1994), which enforces the theoretical basis of the model. To reduce the strong Gibbs oscillations that would arise at the interface if adopting a classic spectral cut-off filter, a Gaussian filter is introduced in conjunction with the LASD model, which has the desirable property of being of compact support in both physical and wavenumber space (Tseng et al. 2006).

#### 2.1.2 Discrete Forcing Immersed Boundary Method

### 2.2 Site Description and Instrumentation

During BUBBLE, a 32-m high tower was deployed inside the 13-m wide “Sperrstrasse” street canyon in Basel, Switzerland \((47^{\circ }33^{\prime }57.20^{\prime \prime }\mathrm {N}, 7^{\circ }35^{\prime }48.80^{\prime \prime }\mathrm {E}, \mathrm {WGS}\ 84)\), as displayed in Fig. 1. The orientation of the street canyon is along the axis \(066^{\circ }\)–\(246^{\circ }\) (east-north-east to west-south-west), the block where the tower was operated is characterized by a length of \(160 \ \mathrm {m}\), and an average width-to-height ratio of \(\xi _\mathrm{c}/z_\mathrm{h} = 1.0\), where \(\xi _\mathrm{c}\) is the street canyon width and where \(z_\mathrm{h}\) is the mean building height. The tower was placed at the midpoint of the block, \(3 \ \mathrm {m}\) away from the north wall, and equipped with six ultrasonic anemometer-thermometers (sonics, labels \(A-F\) in Table 1), mounted on horizontal booms reaching from the tower into the centre of the street canyon.

Details on the ultrasonic anemometer-thermometer (sonic) instrumentation, label, absolute measurement heights *z*, normalized measurement heights (the normalization scale is the location of the highest sonic), sonic type, sampling frequency \(f \ (\mathrm {Hz})\)

Label | \(z \ (\mathrm {m})\) | \(z/z_\mathrm{t}\) | Instrument type | \(f \ (\mathrm {hz})\) |
---|---|---|---|---|

| 3.6 | 0.11 | Gill R2 Omnidirectional | 20.8 |

| 11.3 | 0.35 | Gill R2 Omnidirectional | 20.8 |

| 14.7 | 0.46 | Gill R2 Omnidirectional | 20.8 |

| 17.9 | 0.56 | Gill R2 Omnidirectional | 20.8 |

| 22.4 | 0.7 | Gill R2 Asymmetric | 20.8 |

| 31.7 | 1 | Gill HS | 20.0 |

### 2.3 The Urban Canopy Dataset

*x*,

*y*,

*z*), as in Fig. 1. The probability density function (

*p.d.f.*) of roof heights is characterized by a trimodal distribution (see left plot in Fig. 2) with modes at \(z \approx 4.5 \ \mathrm {m}\) (Mo\(_1\)), \(z \approx 17.5 \ \mathrm {m}\) (Mo\(_2\)) and \(z \approx 22.5 \ \mathrm {m}\) (Mo\(_3\)). The mean roof height \(z_\mathrm{h}\) is \(15.3 \ \mathrm {m}\) and the variance of the roof height is \(6.4 \ \mathrm {m}\). The first mode Mo\(_1\) corresponds to one-storey buildings in the backyards (garages, commercial buildings, etc.), the second mode Mo\(_2\) is related to the the main residential (attached) buildings that line streets and enclose courtyards, whereas the third mode Mo\(_3\) is linked to building N.6 in Fig. 1, whose large surface has a significant impact on the

*p*.

*d*.

*f*of the surface heights.

### 2.4 Processing of the Profile Tower Dataset

*u*,

*v*,

*w*and virtual acoustic temperature \(\theta \) were continuously recorded at all six levels simultaneously from December 1 2001 to July 15 2002. Data acquisition systems and quality control procedures including wind-tunnel calibrations of the instruments are described and documented in Christen (2005);

*u*,

*v*and

*w*statistical moments up to order three were calculated and stored for blocks over 5 min. No filtering was applied to the signal nor standard de-trending, to ensure energy conservation and enable vertical gradients of the state variables to be properly computed. To provide data for comparison with pressure-driven simulations the following processing is further performed:

- 1.
Data are averaged in blocks of 30 min.

- 2.
Data are selectively sampled from the year-round dataset based on the wind direction computed at the tower top sensor. Only 30-min blocks characterized by an approaching wind direction of \(\alpha = 66^{\circ } \pm 10^{\circ }\) (along-canyon regime) and of \(\alpha = 156^{\circ } \pm 10^{\circ }\) (across-canyon regime) throughout the \(6 \times 5\)-min intervals are kept.

- 3.
In order to eliminate the influence of thermal stability, the periods are further filtered based on the classic stability parameter \(\zeta = (z - z_\mathrm{d})/L\) (Stull 1988), where

*L*is the Obukhov length \((L = \theta u^2_{\tau } / [\kappa g \theta _*])\) calculated with both friction velocity \(u_{\tau }\) and scaling temperature \(\theta _*\) measured at the tower top. Only periods characterized by near-neutral stability are retained, \(-0.1 \le \zeta \le +0.1\). The displacement height is computed as \(z_\mathrm{d}=(2/3) z_\mathrm{h}\), in the typical range suggested for high-density urban roughness elements (Grimmond and Oke 1999). - 4.
Cases characterized by \(u_{\tau } \le 0.15 \ \mathrm {m\, s^{-1}}\) at tower top are excluded from the analysis.

Geometrical and numerical parameters for the LES runs

ID | \( z_0 \ \mathrm {(m)}\) | \(\alpha \) (\(^\circ \)) | SGS model |
---|---|---|---|

| \(\varDelta /15\) | 66 | SMAG |

| \(\varDelta /30\) | 66 | SMAG |

| \(\varDelta /15\) | 156 | SMAG |

| \(\varDelta /30\) | 156 | SMAG |

| \(\varDelta /15\) | 66 | LASD |

| \(\varDelta /30\) | 66 | LASD |

| \(\varDelta /15\) | 156 | LASD |

| \(\varDelta /30\) | 156 | LASD |

### 2.5 Set-up of Simulations

Simulations are performed over a regular domain, of size \(L_x \times L_y \times L_z = 512 \times 512 \times 160\), (horizontally) centered at the tower locations \((x_\mathrm{t},y_\mathrm{t})\) and discretized with a 1-m stencil in the three coordinate directions (*x*, *y*, *z*). Numerical parameters for each run are summarized in Table 2. Two directions of the incoming flow are considered, \(\alpha = 66^{\circ }\) and \(\alpha = 156^{\circ }\), which correspond to an along-canyon and across-canyon wind regime respectively. The flow is forced by imposing a constant pressure gradient \(\partial _x p_{\infty } /\rho \), which, in conjunction with lateral periodic boundary conditions, defines a friction velocity \(u_{\tau } = \sqrt{(\delta -z_\mathrm{d})\partial _x p_{\infty }/\rho } \approx 1.23 \text { m s}^{-1}\), making the system independent of Reynolds number effects (fully rough flow regime). Under such conditions it is possible to scale the solution throughout the boundary layer with a characteristic velocity *U*, since molecular diffusion is negligible. The relatively homogeneous integral morphometric statistics and building height in the neighbourhood justifies the pressure forcing in conjunction with lateral periodic boundary conditions (the main surface transition occurs at \({\approx }700 \ \mathrm {m}\) in the radial direction from the tower location). Domain size was chosen based on a sensitivity study (not shown). The hydrodynamic roughness length \(z_0\), defining the surface roughness, is not known a priori; here, \(z_0\) is defined based on a Nyquist-Shannon representation criterion (Shannon 1949): adopting a reference grid stencil \(\varDelta \), the smallest flow/surface feature that can be represented through the Fourier partial sums is \(k_{\varDelta } = 2\varDelta \), whereas all scales smaller than \(k_{\varDelta }\) need to be modelled. Since \(z_0 = 0.033 k_\mathrm{s}\), where \(k_\mathrm{s}\) is the equivalent Nikuradse sand grain roughness, and given that \(k_\mathrm{s} \rightarrow k\) in the limit of negligible viscous effects (*k* is the height of the considered roughness element), we have that \(z_0 = 0.033 k_{\varDelta } \approx \varDelta /15\). To account for variations in the solution due to the \(z_0\) parameter, \(z_0= \varDelta / 30\) is also considered. To reduce the computational time required to reach a dynamic equilibrium, the initial velocity field for each simulation is imposed through interpolation from results of a run at coarser resolution (twice as coarse in each coordinate direction). Equations are integrated in time for 480 non-dimensional time units \(T = z_\mathrm{h}/u_{\tau }\) (\({\approx }2 \ \mathrm {h}\) in dimensional time) in the coarser grid, before being used as the initial condition for the finer grid, where they are further integrated for 250*T*. A time 100*T* is required in order to achieve statistical stationarity in the velocity field and 150*T* is used to compute statistics, which ensures convergence of first- and second-order moments to the corresponding expected values. To further reduce computational costs the \(\delta /z_\mathrm{h} \gtrapprox 50\) requirement (Jimenez 2004) is here sacrificed; simulations are characterized by \(\delta /z_\mathrm{h} = 10.6\). Roughness has a great influence on turbulence up to \({z/z_\mathrm{h} \approx \min (1+D/z_\mathrm{h},5)}\), where *D* is the separation distance between nearest-neighbour roughness elements (Raupach and Thom 1981; Jimenez 2004). Assuming the top of the RSL to be located at \(z/z_\mathrm{h}=5\) implies that the current geometry does not allow an ISL to survive. The limited \(\delta /z_\mathrm{h}\) in the proposed study might be justified by considering that the focus is on the dynamics within the RSL. In these regions turbulence is expected to be strongly affected by the morphology of the roughness elements and only in a minor part by the dynamics of the logarithmic and outer layers (Anderson 2016).

## 3 Results and Discussion

### 3.1 Properties of the Instantaneous Velocity Field

The relatively high variance characterizing the distribution of roof heights (\(\sigma _{z_\mathrm{h}}/z_\mathrm{h} = 0.42\)) causes a transitional behaviour between skimming flow and wake interference flow (see definition of flow regimes in Oke 1988), despite the high value of the plan-area fraction covered by buildings (see Fig. 2). The lower part of the RSL (\(z / z_\mathrm{h} < 2\)) is mainly composed of wake and of non-wake regions (Böhm et al. 2013), whereas higher up in the boundary layer the flow organizes itself into a set of relatively high-speed and low-speed streamwise elongated streaks.

### 3.2 Mean Flow Velocity

*x*,

*y*), including the interior of the roughness elements.

Figures 4 and 5 compare DA and locally-sampled (i.e., extracted from the LES at the tower location) time-averaged \(\tilde{u}\) and \(\tilde{w}\), against the corresponding mean tower-measured data for the two considered approaching wind directions (\(\alpha = 66^{\circ }\) and \(\alpha = 156^{\circ }\)). The locally-sampled time-averaged LES velocity component \(\overline{\tilde{u}}^*(x_\mathrm{t}^*,y_\mathrm{t}^*,z^*)\) compares well against \( \overline{u}_{\mathrm{tower}}^*\) for both wind directions and all heights. Locally sampled and DA LES results are characterized by a modest standard deviation (shaded regions in the LES profiles) throughout the RSL, underlying the limited influence of both \(z_0\) and the SGS closure model in this region of the flow. The relatively larger standard deviation characterizing \(\overline{\tilde{u}}^*(x_\mathrm{t}^*,y_\mathrm{t}^*,z^*)\) in the along-canyon wind regime (\(\alpha = 66^{\circ }\)) is mainly due to variation of the dissipation rates across SGS closures. The component \(\overline{\tilde{w}}^*(x_\mathrm{t}^*,y_\mathrm{t}^*,z^*)\) also compares well against the corresponding \( \overline{w}_{\mathrm{tower}}^*\) for both along-canyon (\(\alpha = 66^{\circ }\)) and across-canyon \((\alpha = 156^{\circ })\) wind directions, as displayed in Fig. 5. Flow approaching from \(\alpha = 156^{\circ }\) leads to a convergence of the flow in the along-canyon direction, causing a local updraft at the tower location, as apparent in Fig. 6. This behaviour is in agreement with both tower measurements and wind-tunnel results of Feddersen (2005). Further, the lack of a recirculation region for flow approaching from \(\alpha = 156^{\circ }\) (across-canyon regime) is consistent Kastner-Klein and Rotach (2004), where street canyons characterized by pitched roofs were connected with no recirculation regions. Flow approaching from \(\alpha = 66^{\circ }\) leads to the formation of a long recirculation bubble down wind of building 8 (see Fig. 1), which extends to the tower location (see Fig. 6), hence influencing local statistics. This underlines the strong dependency of the system on the horizontal extension of the computational domain, which should be as large as possible, in particular in the stream wise direction \(L_x\), to account for upwind buildings and given the strong correlation of the flow in this coordinate direction. The high variance of \(\overline{w}_{\mathrm{tower}}^*\) in Fig. 5 is mainly related to, (a) the small magnitude of mean vertical winds speed, (b) the flow distortion caused by instrument heads in the vertical direction, and (c) the inability to perfectly align sensor heads (no streamline rotation was performed) (Aubinet et al. 2012). DA profiles are characterized by an inflection point \(z_{\gamma }\) for both incoming wind directions, suggesting the presence of a mixing-layer type regime, similar to that observed in flow over a uniform strip canopy (Raupach et al. 1991) and in flow over vegetation canopy (Raupach et al. 1996; Hout et al. 2007). Note however that both studies, characterized by roughness of uniform height, identified the inflection point at \(z_\mathrm{h}\) (i.e. \(z_{\gamma } = z_\mathrm{h}\)). In the current study, the inflection point \(z_{\gamma }\) coincides with an effective building height \(z_e\) (Christen 2005), which can be defined as the averaged surface height, if only buildings higher than \(12 \ \mathrm {m}\) are considered. Introducing an effective building height \(z_e\) allows description of \(z_{\gamma }\) as a function of the surface height distribution, and is justified given that the majority of low buildings in the backyards that make up Mo\(_1\) (see Fig. 2) do not influence the flow. Further, relating \(z_{\gamma }\) to \(z_\mathrm{e}\) allows to recover the limiting behaviour \(\lim _{\sigma _{z_\mathrm{h}} \rightarrow 0}{z_{\gamma } = z_\mathrm{e} = z_\mathrm{h}}\) (i.e. when the canopy is characterized by elements of uniform height, the inflection point corresponds to the mean building height). The relatively high location for the inflection point is due to the presence of strong shear layers that separate from the higher roofs and resist penetration by large structures from above (Coceal et al. 2006), thus providing a natural separation layer between high-speed and low-speed regions. Local profiles are very dependent on the specific features of the urban morphology throughout the RSL, and are therefore not representative of DA quantities. For the along-canyon regime (\(\alpha = 66^{\circ }\)) locally sampled stream wise velocities \((\overline{\tilde{u}}(x_\mathrm{t},y_\mathrm{t},z))\) depart from their DA counterparts \((\langle \overline{\tilde{u}} \rangle )\) in the RSL, mainly due to the persistence of a streamwise elongated low-speed streak, which is locked at the canyon location. This might partly be favoured by the modest vertical and horizontal extensions of the computational domain, which do not allow a full representation of such large-scale structures. However, a similar behaviour was observed in preliminary tests of flow over a larger domain size \((1536 \times 1536 \times 512 \ \mathrm {m})\) (not shown), which suggests that locking of high-speed and low-speed streaks between high-rise buildings is a typical feature of RSL turbulence, and promotes the use of a local scaling approach to collapse profiles in the RSL.

### 3.3 Momentum Fluxes

*z*plane with the solid interface (the buildings). Integrating Eq. 6 analytically in the interval \(z \in (z_\mathrm{h_\mathrm{max}},\delta ]\), results in

#### 3.3.1 Turbulent Fluxes

#### 3.3.2 Dispersive Fluxes

*z*is justified by the large variance of the surface height distribution \((\sigma _{z_\mathrm{h}} = 0.42 z_\mathrm{h})\). From Fig. 8 it is also clear how dispersive momentum fluxes span a broader range of values when compared against their turbulent counterpart in the RSL, highlighting the strong spatial heterogeneity of such terms and the presence of regions in the UCL where strong contributions to the total momentum flux occur (we were however not able to identify any coherent spatial trend).

#### 3.3.3 Pressure Drag

DA total pressure (or form-induced) drag is the main sink of momentum in the UCL. In such a region this drag decreases approximately linearly with height from its surface value \(\int _{0}^{\delta }{\frac{1}{\rho } \Big \langle \frac{\partial \overline{\tilde{p}}^{\prime \prime }}{\partial x} \Big \rangle }{\mathrm{d}z} \approx u_{\tau }^2\). The total pressure drag is non-zero up to the height of the tallest building \((z/z_\mathrm{h} =4.18)\), but it is of negligible magnitude above \(z/z_\mathrm{h} \approx 1\), when compared against the DA turbulent stresses. As is apparent from Fig. 8, the largest contribution to the form drag arises at the windward side of buildings, where positive horizontal gradients of pressure occur as the flow approaches the facade.

#### 3.3.4 Subgrid-Scale Fluxes

SGS fluxes peak at \(z_{\gamma } = z_\mathrm{e}\), due to the presence of thin shear layers of fine-scale turbulence (see Fig. 8), but represent a minor contribution to the total momentum flux in the vertical direction. It is important to recall that despite the minor role of SGS terms in the momentum balance, variations in SGS closure, and thus in the related dissipation rates, can have a strong impact on the resolved scale features, via the impact of SGS terms on the kinetic energy of the flow. Given that the wall-modelled stresses are also SGS terms, results suggests that when urban-type surface roughness is directly resolved (through e.g. an immersed boundary method algorithm), the solution is not sensitive to the wall model. This is reassuring, given the lack of a universal law-of-the-wall for flows in complex geometries.

### 3.4 Budget of TKE

#### 3.4.1 Turbulent and Wake Kinetic Energy

#### 3.4.2 Production Terms

*p.d.f.*of building heights (see Fig. 2), which can be regarded as a very specific feature of the current set-up, linked to the shear layers separating from building N. 6 in Fig. 1. \(\langle P_\mathrm{w} \rangle \) is the production rate of TKE in the wakes of roughness elements by the interaction of local turbulent stresses and time-averaged strains; in the lower UCL it is approximately constant, positive (WKE converts to TKE) of magnitude \(\langle P_\mathrm{w} \rangle ^* \approx u_{\tau }^3/z_\mathrm{h}\). \(\langle P_\mathrm{w} \rangle \) accounts for over \(50\,\%\) the total production rate of TKE in the UCL, and is therefore non-negligible. A previous study of flow over uniform strip canopy (Raupach et al. 1991) found \(\langle P_\mathrm{w} \rangle \) to increase linearly in the canopy, reach a maxima \(\langle P_\mathrm{w} \rangle \approx \langle P_\mathrm{s} \rangle \) at \(z_\mathrm{h} = z_{\gamma }\), and rapidly decrease to zero in the lower RSL. In experimental and numerical studies of flow over gravel beds (Mignot et al. 2009; Yuan and Piomelli 2014) the magnitude of \(\langle P_\mathrm{w} \rangle \) was found to be less than \(5\,\%\) of \(\langle P_\mathrm{s} \rangle \) (based however on a superficial averaging). \(\langle P_\mathrm{w} \rangle \) thus seems to strongly vary as a function of the roughness properties. Our results suggests that in flows over realistic urban canopies the presence of street canyons aligned with the mean flow, open areas and variable building geometries tends to increase \(\langle P_\mathrm{w} \rangle \) in the lower UCL (\(z^* \lessapprox 0.5\)), when compared to results of flow over regular canopy (see for example Raupach et al. (1991)). The additional form-induced production term \(\langle P_\mathrm{m} \rangle \) is non-zero only in the vicinity of the inflection layer \(z_{\gamma }\), where it accounts for \(16\,\%\) the magnitude of \(\langle P_\mathrm{s} \rangle \).

Percentage contribution of production, dissipation and transport terms to the total source and sink rate of TKE for the considered layers

Layer | Production | | Transport |
---|---|---|---|

UCL \((0 < z < z_\mathrm{h} )\) | \(60\,\%\,(+)^*\) | \(100\,\%\,(-)^*\) | \(40\,\%\, (+)\) |

Upper RSL \((z_\mathrm{h} < z < 5 z_\mathrm{h} )\) | \(100\,\%\, (+)\) | \(88\,\%\,(-)\) | \(12\,\%\,(-)\) |

ISL \((z > 5 z_\mathrm{h} )\) | \(95\,\%\, (+)\) | \(100\,\%\,(-)\) | \(5\,\%\, (+)\) |

#### 3.4.3 Transport Terms

Turbulent transport terms are compared against tower measurements in Fig. 12. Numerical results and measurements are in good agreement, apart from an overshoot of the numerical \(T_\mathrm{t}(x_{t},y_{t},z)\) in the across-canyon regime at the height of sonic *E*, suggesting higher resolution might be necessary in order to properly describe the small scale turbulence characterizing the thin shear layers that separate from the roofs of buildings (recall that the current grid stencil is 1 m). Note that in this specific case, DA profiles are in qualitative agreement with data from the same tower, and an additional tower (not shown), operated under a much wider range of stabilities during BUBBLE (Christen et al. 2009).

#### 3.4.4 Dissipation and Residual Terms

Normalized horizontal standard deviation (\(\sigma ^*\)) for selected statistics

Quantity | \(0 \le z/z_\mathrm{h} < 0.5\) | \(0.5 \le z/z_\mathrm{h} < 1 \) | \( 1 \le z/z_\mathrm{h} < 3\) | \( 3 \le z/z_\mathrm{h} < 5 \) |
---|---|---|---|---|

\(\sigma ^*_\mathrm{TKE}\) | 0.6 | 0.5 | 0.25 | 0.2 |

\(\sigma ^*_{u^{\prime }w^{\prime }}\) | 2.3 | 1.2 | 0.45 | 0.35 |

\(\sigma ^*_{P_\mathrm{s}}\) | 70 | 10 | 5 | 6 |

\(\sigma ^*_{T_\mathrm{t}}\) | 20 | 43 | 37 | 120 |

\(\sigma ^*_{-\epsilon }\) | 4 | 3.5 | 4 | 4 |

### 3.5 On the Representativeness of Local Measurements in the RSL

As stated in Sect. 1, field-studies are usually sampling the flow at few points in space, and therefore cannot account for its spatial variability and for dispersive contributions. The very nature of RSL turbulence hence questions the usage of point measurements as surrogate of horizontally averaged quantities in such regions, as underlined in Rotach (1993a, b) and Christen et al. (2009). Unfortunately, the vast range of urban geometries limits the scope of any investigation aiming at defining confidence bounds for locally measured quantities. Without ascribing generality to the proposed results, we here summarize the spatial variability of turbulent statistics and the contribution of dispersive terms in the RSL for the considered study. Such information is of use to ensure the representativeness of local measurements on sites.

*z*intervals. \(\sigma ^*\) is related to sampling at different horizontal locations in space (within the fluid only) and is defined as

*N*denotes the number of collocation nodes in a horizontal layer considering fluid areas only (i.e. not within buildings). Quantities \(\sigma _\mathrm{TKE}^*\) and \(\sigma _{u^{\prime }w^{\prime }}\) are characterized by a monotonic decrease from their surface value, but remain finite throughout the UCL and RSL. Based on current results, local measurements of TKE and \(u^{\prime }w^{\prime }\) should account for a standard deviation up to about 60 and \(230\,\%\) the magnitude of the corresponding sampled mean in the (lower) UCL. The same values decrease to 25 and \(45\,\%\) respectively in the above-UCL regions \((z > z_\mathrm{h})\). Note that the proposed percentages are in qualitative agreement with results displayed in Figs. 7 and 9. Table 4 highlights a remarkable spatial variability of \(P_\mathrm{s}\) and \(T_\mathrm{t}\) the RSL, tightly related to the strength of shear layers that characterized the flow in such regions (as apparent from Fig. 13). Sensor deployment within the RSL should therefore be performed avoiding such high shear rate regions, which would otherwise cause an overestimation of the measured \(P_\mathrm{s}\) and \(T_\mathrm{t}\), relative to their spatial mean. This is confirmed by results in Figs. 11 and 12: in the across-canyon wind regime (\(\alpha =156^{\circ }\)) sonics

*C*,

*D*,

*E*(see Table 1) are sampling within one of such shear layers, and the resulting values of \(P_\mathrm{s}\) and \(T_\mathrm{t}\) are clearly not representative of their spatially averaged value. Note that the large \(\sigma ^*_{T_\mathrm{t}}\) in the upper RSL (\(3 \le z/z_\mathrm{h} < 5\)) is likely related to the negligible magnitude of \(T_\mathrm{t}\) in such layer. Besides, the magnitude of the computed coefficients in the RSL is likely amplified by the presence of a relatively taller building (building N.7 in Fig. 1), whose effects on the resulting \(\sigma ^*\) remain significant up to a height of \(z/z_\mathrm{h} \approx 5\).

Ratio of dispersive to Reynolds contributions (\(\xi \)) for selected statistics

Quantity | \(0 \le z/z_\mathrm{h} < 0.5\) | \(0.5 \le z/z_\mathrm{h} < 1 \) | \( 1 \le z/z_\mathrm{h} < 3\) | \( 3 \le z/z_\mathrm{h} < 5 \) |
---|---|---|---|---|

\(\xi _\mathrm{TKE}\) | 1.3 | 0.9 | 0.3 | 0.2 |

\(\xi _{u^{\prime }w^{\prime }}\) | 0.6 | 0.8 | 0.2 | 0.3 |

\(\xi _{P_\mathrm{s}}\) | 14 | 1.5 | 0.3 | 0.4 |

\(\xi _{T_\mathrm{t}}\) | 0.9 | 0.9 | 0.5 | 0.5 |

Overall, Tables 4 and 5 suggest that point-wise measurements of TKE and \(u^{\prime }w^{\prime }\) are equally biased by the spatial heterogeneity of the flow statistics and by the presence of additional dispersive contribution from the mean flow. Conversely, local sampling of TKE budget terms is largely biased by their spatial heterogeneity, which despite the remarkable magnitude of the \(\sigma ^*\) parameters, does not lead to significant contributions from the mean flow (exemplified by the relatively modest \(\xi \) values).

## 4 Conclusions

A characterization of mean flow and turbulence in the RSL of a realistic urban canopy, representing a subset of the city of Basel in Switzerland, has been performed via a series of large-eddy simulations (LES) and results have been compared to direct tower measurements from a long-term field campaign. First-order and higher order statistics compare well against tower measurements, confirming that LES in conjunction with the immersed boundary method is a valuable tool for the simulation of flow and dispersion over realistic urban surfaces. Double averaged numerical profiles are not sensitive to variations in both the subgrid-scale (SGS) model and the hydrodynamic roughness length \((z_0)\) parameter, given that form drag represents a significant percentage of the total surface drag, and is well resolved through the IBM. Double-averaged velocity profiles are characterized by an inflection point \(z_{\gamma }\), located above the mean building height \(z_\mathrm{h}\), highlighting the presence of a mixing-layer type flow regime. Double-averaged Reynolds fluxes and double averaged turbulent kinetic energy (TKE) peak above \(z_{\gamma }\), in agreement with results from studies of flow over simplified urban-like surfaces. TKE is significant in the urban canopy layer (UCL), when compared against results of flow over gravel beds and over regular / random arrays of cubes, mainly due to the presence of flow-aligned street canyons, open areas and a variable building height, which strongly increase the strength of both mean kinetic energy and TKE in such regions. Further, dispersive momentum fluxes and dispersive production and transport of TKE are found to be non-negligible in the UCL, and of the same order of magnitude of their Reynolds counterparts. TKE is primarily produced at \(z_{\gamma }\) by shear, and is transported down into the cavities of the urban canopy (street canyons, backyards) by turbulent and dispersive transport terms, which share similar magnitudes. Transport terms are non-negligible throughout the RSL. They are of negative sign and contribute to about \(12\,\%\) the total variation rate of TKE in the upper RSL (\(z_\mathrm{h} < z < 5 z_\mathrm{h}\)), whereas they are of highest significance in the UCL (\(0 < z < z_\mathrm{h}\)), where they are of positive sign and contribute to about \(40\,\%\) the local variation rate of TKE. Wake production is roughly constant up to \(z_{\gamma }\) and of non-negligible magnitude \((\langle P_\mathrm{w} \rangle ^* \approx u_{\tau }^3/z_\mathrm{h})\), contributing up to \(50\,\%\) the total TKE production rate in the UCL. Further, pressure transport is found to be a significant source of TKE in the near-wall regions, in agreement with previous findings of flow over vegetation canopy and flow over gravel beds. The spatial heterogeneity and the dispersive contribution of selected flow quantities are summarized for reference intervals in the RSL. Results highlight how RSL tower measurements can be severely biased because of the spatial heterogeneity of the flow. Further, tower measurements cannot be used to quantify all terms in a horizontally-averaged view: dispersive terms are important in a real canopy. This also means that one-dimensional exchange models in urban canopy parametrizations relying commonly solely on turbulent fluxes will underestimate the exchange. Dispersive fluxes should therefore be considered in the exchange computation of future urban canopy parametrization schemes.

## Notes

### Acknowledgments

This work was supported by National Science Foundation and by a grant from the Swiss National Supercomputing Centre (CSCS) under projects ID s404 and s599. CM acknowledges the hospitality and support of EPFL during a visit during 2013, and MBP is grateful for the support provided by the NSERC Discovery Grant.

## Supplementary material

Supplementary material 1 (mp4 5118 KB)

Supplementary material 2 (mp4 3341 KB)

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