Large-Eddy-Simulation Study of the Effects of Building-Height Variability on Turbulent Flows over an Actual Urban Area

Abstract

Large-eddy simulation (LES) is used to investigate the effects of building-height variability on turbulent flows over an actual urban area, the city of Kyoto, which is reproduced using a 2-m resolution digital surface dataset. Comparison of the morphological characteristics of Kyoto with those of European, North American, and other Japanese cities indicates a similarity to European cities but with more variable building heights. The performance of the LES model is validated and found to be consistent with turbulence observations obtained from a meteorological tower and from Doppler lidar. We conducted the following two numerical experiments: a control experiment using Kyoto buildings, and a sensitivity experiment in which all the building heights are set to the average height over the computational region $h_{all}$. The difference of Reynolds stress at height $z=2.5h_{all}$ between the control and sensitivity experiments is found to increase with the increase in the plan-area index ($\lambda _p$) for $\lambda _p > 0.32$. Thus, values of $\lambda _p\approx 0.3$ can be regarded as a threshold for distinguishing the effects of building-height variability. The quadrant analysis reveals that sweeps contribute to the increase in the Reynolds stress in the control experiment at a height $z= 2.5h_{all}$. The exuberance in the control experiment at height $z=0.5h_{all}$ is found to decrease with increase in the building-height variability. Although the extreme momentum flux at height $z=2.5h_{all}$ in the control experiment appears around buildings, it contributes little to the total Reynolds stress and is not associated with coherent motions.

1 Introduction

Atmospheric processes over urban areas are affected not only by meteorological disturbances, such as thunderstorms, fronts, and cyclones, but also by the roughness and thermal effects of buildings and man-made structures. The geometrical features of buildings and structures determine the roughness effects of an urban area, while human activities and the material characteristics of buildings play a role in defining the thermal effects of such areas. The complex geometrical nature of urban surfaces results in highly complex turbulent flows. To properly understand the physical processes of momentum and heat transfer in urban areas and to develop parametrizations for urban environments in numerical weather prediction models, it is important to reveal relationships between the effects of actual urban buildings and turbulent flows.

The characteristics of turbulent flows over urban surfaces have been examined in numerous previous studies. Oke (1988) categorized the airflow over roughness obstacles as a function of obstacle density as isolated flows, wake-interference flows, and skimming flows. Macdonald et al. (1998) derived a theoretical relation for the aerodynamic roughness length $z_0$ and displacement height d for flows over roughness blocks. While these studies examined turbulent flows over roughness blocks with constant height and regular distribution, the recent focus has shifted to the effects on turbulent flows of roughness blocks with variable height and inhomogeneous arrangement. Wind-tunnel experiments (Cheng and Castro 2002) have demonstrated that the roughness sublayers over block arrays with random height are thicker than those over uniform-height arrays. Xie et al. (2008) conducted a large-eddy simulation (LES) of turbulent flows over block arrays with random height, and found that the tall blocks significantly contribute to the total drag of such arrays. Nakayama et al. (2011) performed LES investigations over building arrays with different height variability and found that the vertical profiles of the mean velocity and Reynolds stress depend significantly on the building-height variability. Zaki et al. (2011) performed wind-tunnel experiments with block arrays of buildings with variable height distributed randomly, and showed that the drag coefficient $C_d$ increases with the building density and the standard deviation of the building height for high building densities. Numerical simulations of plume dispersion over urban surfaces have revealed that the turbulence is significantly affected by the source location and wind direction because of the strong dependence on the building height and distribution. (Xie and Castro 2009; Xie 2011; Nakayama et al. 2016). The parametrizations of $z_0$ and d have been improved by taking into account roughness parameters associated with actual urban buildings, such as the maximum, standard deviation, and skewness of the building height (Nakayama et al. 2011; Kanda et al. 2013; Zhu et al. 2017). Giometto et al. (2016) suggested that the dispersive flux derived from spatial variations of temporal mean flows around buildings should be considered so as to improve conventional urban-canopy parametrizations.

To fully understand the effects of roughness obstacles on turbulent flows, it is helpful to investigate the relationships between turbulent organized structures and obstacles, because organized structures are associated with downwards momentum transfer in the form of ejection and sweep events based on a quadrant analysis for the turbulent momentum flux. The results of wind-tunnel experiments on flows over rough surfaces (Raupach 1981) indicate that sweeps are dominant for the total momentum flux near surfaces, and that the contribution of ejections to the momentum flux increases with height. Studies in which turbulence was observed over actual urban areas have revealed the characteristics of momentum transfer and coherent motion. Thus, Oikawa and Meng (1995) observed turbulent structures associated with ejections and sweeps over an urban area, and found that turbulent structures correlate with heat transfer within and above the urban canopy. Christen et al. (2007) analyzed field experimental data obtained from sonic-anemometer measurements within and above a street canyon in Basel, Switzerland, and found that sweeps are mostly dominant up to a height of approximately twice the average building height in a street canyon. Numerical simulations of flows over building arrays have revealed the spatial characteristics of turbulent organized structures. Kanda et al. (2004) carried out LES investigations of turbulent flows over uniform-height block arrays to investigate turbulent organized structures over such arrays. They found low-speed streaks and streamwise vortices similar to those in flows over flat-wall boundary layers. Kanda (2006) indicated that streak structures are a common feature over various types of block arrays. Using direct numerical simulations, Coceal et al. (2007a, b) revealed that hairpin vortices associated with ejections and sweeps are generated over uniform block arrays, and that the low-speed streaks identified above such arrays are composed of large numbers of hairpin vortices aligned in the streamwise direction. Park et al. (2015) used LES results to analyze turbulent-flow structures over an actual urban area in Seoul, Korea, and showed that turbulent structures behind high-rise buildings are characterized by streamwise vortices with strong ejections. They focused on small regions containing high-rise buildings, and demonstrated the significant influence of high-rise buildings on wake flows. The majority of studies presented thus far have focused on the characteristics of turbulent flows over idealized or specific buildings, while only a few have examined the urban-scale effects on the characteristics of turbulent momentum transfer produced by the complex geometrical features of actual urban surfaces.

The geometrical characteristics of actual urban surfaces can be reproduced from digital surface datasets. Ratti et al. (2002) calculated the roughness parameters of North American and European cities, and found that parameters differ significantly by city. Bou-Zeid et al. (2009) indicated that turbulent flows are dependent on the building representation over the actual urban surface. To understand the characteristics of turbulent flows over urban areas, it is therefore important to use the geometry of actual buildings in simulations and experiments.

We investigate here the effects of building-height variability in an actual urban area on turbulent flows at an urban scale, focusing on the airflow within and above an urban-canopy layer, where turbulent flows are strongly influenced by individual buildings.

We simulate the turbulent flow over the urban area of Kyoto, which is characterized by the presence of both business districts with high buildings and densely built residential districts. Furthermore, observations from a meteorological tower owned by Kyoto University and located in the southern part of the city can be used for the validation of simulations. In Sect. 2, the building morphological characteristics of Kyoto are evaluated using roughness parameters. The details of our LES model are described in Sect. 3. The study area of the LES investigation is defined to include the meteorological tower site at which turbulence was measured by a sonic anemometer and Doppler lidar, so that LES results may be compared with the observations (see Sect. 4). Along with a control simulation, we conduct a sensitivity test assuming a constant building height to reveal the effects of building height–height variability, with the differences between the control and sensitivity experiments examined in Sect. 5. Finally, Sect. 6 gives conclusions.

2 Building Morphological Characteristics of Kyoto

Our study area covered both business districts and suburban areas in Kyoto, with Fig. 1 showing the area of interest in Kyoto, which extends 11 km in a north–south direction and by 2 km in an east–west direction. A digital surface model (Kokusai Kogyo Co., Ltd.) was used to reproduce the actual urban buildings within a numerical model. The original 2-m-resolution data are smoothed and converted to a 4-m resolution, which is used as the horizontal grid spacing of the numerical experiments as described in Sect. 3.2.

Fig. 1

The study area in which the large-eddy simulations and observations were carried out is indicated by the red box. The observational site of the Disaster Prevention Research Institute, Kyoto University, is indicated by the white circle. The white arrow indicates the streamwise wind direction. The satellite picture is taken from Google Earth

Fig. 2

a Distribution of building and structure heights in the analysis region of Kyoto. b Frequency distribution of building heights in the analysis region. The black bar indicates the frequency distribution of buildings in the overall region, while the grey and hatched bars indicate the frequency distributions of buildings in the regions with $x = 0{-}4$ km and with $x = 7{-}11$ km, respectively

Figure 2a shows the height of the actual buildings in the analysis area. The north–south and west–east directions are referred to as the x and y directions, respectively, with the region $x =0{-}4$ km corresponding to the city centre of Kyoto. The heights of almost all buildings in the region are up to 50 m, and there are no high-rise building clusters of the type seen in the centre of Tokyo. The region for $x = 7{-}11$ km is primarily occupied by suburban areas and rivers.

The difference between the building heights over these two regions is clearly indicated in Fig. 2, which shows the frequency distributions of building heights over the entire analysis area and in the $x = 0 {-} 4$ km and $x = 7 {-} 11$ km regions. In calculating the frequency distributions, all buildings are defined as having heights of at least 1 m to distinguish between the buildings and the ground. It is seen that most of the buildings taller than 25 m are located in the former region.

Fig. 3

a $\lambda _p$ calculated for 1 km by 1 km areas over the analysis region. b Roughness parameters calculated for 1 km by 1 km areas over the analysis region. In each box, the first row is $H_{ave}$ (m), the second is $\sigma _H$ (m), the third is $\lambda _p$, and the fourth is $\lambda _{f}$. Scatter plots c between $\lambda _p$ and $\lambda _{f}$ and d between $\sigma _H$ and $H_{ave}$. The black lines in c, d the empirical relationships derived from Tokyo and Nagoya, respectively, by Kanda et al. (2013). The values of Salt Lake City and Los Angeles in North America and London, Toulouse, and Berlin in Europe, as indicated by the lower legend, are obtained from Ratti et al. (2002)

To quantitatively indicate the morphological characteristics of buildings in Kyoto, we use roughness parameters such as the average building height $H_{ave}$, the standard deviation of the building height $\sigma _H$, the plan-area index $\lambda _p$ (the ratio of the plan area occupied by buildings to the total surface area), and the frontal-area index $\lambda _{f}$ (the ratio of the frontal area of buildings to the total surface area). These parameters are calculated for each 1 km by 1 km area following the analysis of Kanda et al. (2013). Figure 3a shows $\lambda _p$ calculated in the areas of 1 km by 1 km for the buildings shown in Fig. 2a, with the values of the roughness parameters in the 1 km by 1 km areas summarized in Fig. 3b. The average values of $H_{ave}$, $\sigma _H$, $\lambda _p$, and $\lambda _{f}$ over the $x = 0 {-} 4$ km region are 10.8 m, 7 m, 0.41, and 0.25, respectively, while the corresponding averages over $x =7 {-} 11$ km are 9.8 m, 5.3 m, 0.2, and 0.16, respectively. Thus, the $x = 0 {-} 4$ km region is more densely built than is the $x =7 {-} 11$ km region. Using building data from Tokyo and Nagoya, Japan, Kanda et al. (2013) derived the following empirical relationships between $\lambda _p$ and $\lambda _{f}$, and between $H_{ave}$ and $\sigma _H$ (units of m),

$$\begin{aligned} \lambda _{f}= & {} 1.42\lambda _{p}^2 + 0.4\lambda _{p}, \end{aligned}$$

(1)

$$\begin{aligned} \sigma _H= & {} 1.05H_{ave}-3.7. \end{aligned}$$

(2)

with Fig. 3c, d indicating the respective relationships between $\lambda _{p}$ and $\lambda _{f}$, and between $H_{ave}$ and $\sigma _H$, based on the data given in Fig. 3b. Also shown in the panels are the empirical relationships of Kanda et al. (2013) and the data for North American and European cities found in Ratti et al. (2002). For $\lambda _p > 0.3$, the $\lambda _{f}$ values for Kyoto tend to be smaller than in the empirical profile. This feature of Kyoto appears to be similar to those seen in European cities, and indicates that the fraction of high buildings in Kyoto is limited relative to those in major metropolitan cities in Japan and North America. The relationship between $H_{ave}$ and $\sigma _H$ for Kyoto is in good agreement with those of Tokyo and Nagoya, but differs from those of European cities. Finally, the magnitudes of $H_{ave}$ and $\sigma _H$ in Kyoto are smaller than those of Los Angeles by a factor of 5–10.

According to these results, Kyoto can be morphologically characterized as having densely distributed buildings with widely varying heights. The Kyoto dataset was used for the numerical simulations described in the next section.

3 Numerical Model and Experimental Design

3.1 Numerical Model

Our LES model is effectively the same as that used in Nakayama et al. (2011), except that it neglects the molecular viscosity term, and employs a bottom boundary condition based on Monin–Obukhov similarity theory, as described later. In Nakayama et al. (2011), the performance of the LES model reproducing turbulent statistics was validated using data obtained from wind-tunnel experiments; as close agreement was found, the model developed by Nakayama et al. (2011) has subsequently been applied to simulate turbulent flows over actual urban cities. Nakayama et al. (2012) conducted LES investigations of turbulent flows over Tokyo by coupling their model with a mesoscale meteorological model, and found that observed gust factors are accurately reproduced by the model. The model was also used to successfully reproduce the wind speed and direction at the ground in the Fukushima Daiichi Nuclear Power Plant during the Great East Japan Earthquake and its aftermath in March 2011 (Nakayama et al. 2015). Nakayama et al. (2016) further applied their LES model to the simulation of turbulent flows and plume dispersion over Oklahoma City, and showed that the observed characteristics of turbulence and dispersion are reproduced despite the fact that small differences in wind direction caused by the building distribution significantly influenced the plume dispersion. Thus, our LES model has been widely tested and is applicable for the analysis of the turbulent flow over Kyoto.

The LES model solves the filtered continuity and Navier–Stokes equations in Cartesian coordinates with the subgrid-scale stress parametrized by the standard Smagorinsky model (Smagorinsky 1963). The governing equations are

$$\begin{aligned}&\frac{\partial \tilde{u}_i }{\partial x_i}=0, \end{aligned}$$

(3)

$$\begin{aligned}&\frac{\partial \tilde{u}_i }{\partial t} + \tilde{u}_j \frac{\partial \tilde{u}_i }{\partial x_j}= - \frac{1}{\rho }\frac{\partial \tilde{p}^* }{\partial x_i} - \frac{\partial \tau _{ij}}{\partial x_j} + f_i, \end{aligned}$$

(4)

$$\begin{aligned}&\tau _{ij} - \frac{1}{3}\delta _{ij}\tau _{kk} =-2(C_s{\Delta })^2(2\tilde{S}_{ij}\tilde{S}_{ij})^{1/2}\tilde{S}_{ij}, \end{aligned}$$

(5)

$$\begin{aligned}&\tilde{S}_{ij} = \frac{1}{2} \left( \frac{\partial \tilde{u}_i }{\partial x_j}+\frac{\partial \tilde{u}_j }{\partial x_i} \right) , \end{aligned}$$

(6)

where t denotes time, $\tilde{u}_i$ is the filtered air velocity in the direction i, $\tilde{p}^*=\tilde{p}+\frac{1}{3}\rho \tau _{kk}$ is the modified pressure, $\tilde{p}$ is the filtered pressure, $\rho$ is the density of air, $\tau _{ij}$ is the subgrid-scale stress, $\delta _{ij}$ is the Kronecker delta, $\tilde{S}_{ij}$ is the filtered stress tensor, and $f_i$ is the external force exerted by roughness obstacles. The parameter $x_i$ represents the coordinate system, with components $i=1, 2$, and 3 referring to the streamwise (x), spanwise (y) and vertical (z) directions, respectively. In addition, $\Delta = (\Delta _x\Delta _y\Delta _z)^{1/3}$ is the filter width, where $\Delta _x$, $\Delta _y$, and $\Delta _z$ are the streamwise, spanwise, and vertical grid spacings, respectively. The Smagorinsky coefficient $C_s$ is set to 0.14. Note that the viscous term is neglected because our target is the simulation of turbulent flow at high Reynolds number.

The external force $f_i$ is used to simulate the effects of buildings on the flow, for which we employ the feedback forcing by Goldstein et al. (1993) who give

$$\begin{aligned} f_i= & {} \alpha \int ^{t}_{0} u_i(t') \mathrm{d}t' + \beta u_i(t),\quad \alpha<0,\quad \beta <0, \end{aligned}$$

(7)

where $\alpha$ and $\beta$ are negative constants. The stability limit is given by $\Delta t < \frac{- \beta - \sqrt{(\beta ^2-2\alpha k )}}{\alpha }$, where k is a constant of order one. Following Nakayama et al. (2011), these constants are set as $\alpha = -10$, $\beta = -1$, and $k = 1$.

The governing equations are discretized on a staggered-grid system. The velocity and pressure fields are solved using a coupling method based on the marker-and-cell method (Chorin 1967). The successive over-relaxation method is used to solve the Poisson equation for pressure, and the Adams–Bashforth scheme is adopted for the time integration. A second-order accurate, central-differencing scheme is employed for spatial discretization, and the code is parallelized using a Message Passing Interface library to reduce the computational time.

3.2 Experimental Design

The governing equations are numerically solved in two computational domains: the driver region, which features regularly-arrayed obstacles, and the main region, which contains the actual buildings of Kyoto. To ensure the LES flow field is turbulent, a turbulent flow is generated in the driver region and imposed as the inflow at the boundary of the main region. The concept involved in setting the driver and the main regions is demonstrated in Fig. 4. The size of the driver region is 6 km (streamwise) $\times$ 2.4 km (spanwise) $\times$ 1.015 km (vertical), with a grid spacing of 4 m in the horizontal directions, and a grid spacing stretched with increasing altitude from 1 to 16 m in the vertical direction. The total number of grid points is 1500 $\times$ 600 $\times$ 105. In the driver region, there is one rectangular block aligned in the spanwise direction, and an array of roughness blocks staggered with $\lambda _{p}=0.04$. The individual rectangular and roughness block sizes are 50 m $\times$ 2400 m $\times$ 50 m and 16 m $\times$ 16 m $\times$ 10 m, respectively. The purpose of setting the size of the rectangular block is to enhance perturbations near the inlet of the driver region. The $\lambda _{p}$ value chosen for the block array is set to be a little larger than that in Nakayama et al. (2014) to accelerate the generation of turbulence. The height of the blocks is chosen according to the mean building height in the main region.

Fig. 4

Schematic of turbulent flows formed in the driver region and imposed on the main region as the inflow condition

A uniform flow with a velocity magnitude of $5\,\hbox {m s}^{-1}$ is imposed at the inflow boundary of the driver region. The Sommerfeld radiation condition is imposed at the outflow boundary, while a periodic condition is set at the lateral boundaries. At the top boundary, free-slip and zero-speed conditions are imposed for the horizontal and vertical velocity components, respectively and, at the ground, a boundary condition based on Monin–Obukhov similarity theory is employed. The stress at the first vertical grid $\tau _{i3}(x, y, t)$ is calculated as (Stoll and Porté-Agel 2006)

$$\begin{aligned} \tau _{i3}(x,y,t) = - \left[ \frac{\tilde{u}_r(x,y,z_s,t) \kappa }{\mathrm{{ln}}{(z_s/z_0)}}\right] ^2 \frac{\tilde{u}_i(x,y,z_s,t) }{\tilde{u}_r(x,y,z_s,t) }, \end{aligned}$$

(8)

where $\tilde{u}_r(x,y,z_s,t) = [\tilde{u}_1(x,y,z_s,t)^2+\tilde{u}_2(x,y,z_s,t)^2]^{1/2}$ is the instantaneous resolved velocity magnitude, $z_s$ is the altitude at the first vertical grid, $z_0$ is the roughness length, and $\kappa$ is the von Kármán constant. Here, $z_0=0.1$ m (Bou-Zeid et al. 2009) and $\kappa =0.4$.

The ratio of the boundary-layer height $\delta$ of the generated outflow to the roughness block height in the driver region is 27.9, noting that, here, $\delta$ is defined as the height at which the mean streamwise velocity component at the outflow indicates a peak value. In Nakayama et al. (2011), the ratio of $\delta$ to the roughness block height in the driver region is 13. In addition, we confirmed that the vertical profiles of the standard deviation of each velocity component and Reynolds stress are in reasonable agreement with those obtained from wind-tunnel experiments, although the LES results underestimate the spanwise and vertical components and Reynolds-stress values relative to the wind-tunnel results (see Online Resource 1, Figure 1). These results suggest that well-developed, deep turbulent flows are generated in the driver region.

In the main region, the domain size and the total number of grid points are 12 km $\times$ 2.4 km $\times$1.015 km and 3000 $\times$ 600 $\times$ 105, respectively, with the main region including the actual buildings and structures in Kyoto, as shown in Fig. 2a. For computational purposes, we set a buffer area spanning 500 and 200 m in the streamwise and spanwise directions, respectively, surrounding the actual building area in the main region (not shown in Fig. 2a). The streamwise width of the area was determined based on Nakayama et al. (2012), who carried out a LES investigation of the airflow over Tokyo. Whereas Nakayama et al. (2012) did not set a buffer area in the spanwise direction, we decided that a spanwise buffer is necessary to avoid building discontinuities arising from the periodic boundary conditions. In this buffer area, the same roughness blocks used in the driver region are applied to maintain a turbulent flow over roughness surfaces. Note that the coordinates $x = 0$ and $y = 0$ are set to the northern and western boundaries, respectively, of the actual building area in the main region. Correspondingly, the inflow boundary condition provided by the driver region is set at $x = -\,500$ m in the main region. Outside of the inflow boundary, the boundary conditions of the main region are the same as those in the driver region, and all grid spacings are identical to those in the driver region.

Fig. 5

Time series of a streamwise, and b spanwise velocity components at 25-m height observed by the sonic anemometer during periods D1–D4 and simulated by the LES model. Vertical profiles of c mean streamwise velocity component and d Reynolds stress normalized by the mean streamwise velocity component in the observations and the LES results. Note that the profiles in (c) are plotted on logarithmic axes. A line with a slope 1/4 is also plotted for reference. Here, ‘ane’ and ‘dop’ refer to the observations by the anemometer and Doppler lidar, respectively

Hereafter, the simulation using the actual buildings in Kyoto is referred to as the control experiment (CTL). To reveal the effects of building-height variability, we conducted an additional experiment referred to as the uniform experiment (UNI) in which all building heights are set to the average of the actual building heights in the main region ($h_{all} =10.3$ m). The integration time for each of the two experiments is 7200 s, with the results obtained from the last 1800 s used for the analysis of turbulent statistics. In Sect. 4.3, we confirm that the flows were in equilibrium states during this analysis period, as shown Fig. 5. In addition, as seen in Fig. 1 of Online Resource 1, the second-order moments of the inflow profiles are relatively small compared with those of the wind-tunnel experiments, which possibly influences the results presented here. However, as the same inflow condition was applied in both the CTL and UNI experiments, we can assume that any differences in the respective experimental results are unaffected by this issue.

4 Comparison with Observations

4.1 Observational Setting

The observations were made at the Ujigawa Open Laboratory of the Disaster Prevention Research Institute, Kyoto University, during the period from 12 January to 12 February 2016. The laboratory is located in the southern part of Kyoto, and is surrounded by low-rise buildings and structures. The location of the observation site is shown in Fig. 1, which includes a meteorological observation tower of height 55 m. This tower is a unique facility first deployed in 1978 (Nakajima et al. 1979), and is currently one of the few meteorological towers operating in Japan.

A sonic anemometer (DA-600, Kaijo Co.) installed on the tower at a 25-m height measures the three velocity components as well as the air temperature at a 10-Hz sampling rate. The surrounding area up to 500 m north of the tower has only low building heights (< 25 m), enabling the assumption that observations taken by the sonic anemometer are not influenced by the strong wakes of tall buildings.

We also installed a Doppler lidar (WINDCUBE WLS-7, Leosphere) at the ground near the tower, from which we obtained three-component velocity measurements at heights ranging from 40 to 200 m with a 20-m interval at a sampling rate of 1 Hz.

4.2 Data Selection

The observation site was included in the main region assessed in the numerical experiment for the purpose of directly comparing the LES results in the CTL experiment with the observations. As the sonic anemometer installed on the tower faces northwards, we analyzed data for dominant northerly wind directions to minimize the interference from the tower. To extract suitable periods from the observational data, we imposed two criteria for sorting values obtained from the sonic anemometer. First, a northerly flow condition was adopted by classifying 10-min averaged wind directions into 16 classes and extracting periods when northerly wind directions ($348.25^\circ {-}360^\circ , 0^\circ {-}11.25^\circ$) were sustained for at least 30 min. Note that the time period for the analysis of the LES data was also 30 min. Second, a neutrally-stratified condition was chosen based on the Monin–Obukhov stability parameter

$$\begin{aligned} \frac{z}{L}= & {} -\frac{{(g/\overline{T})}{ \overline{w'T'}}}{u_*^3/\kappa z}, \end{aligned}$$

(9)

so that the assumption of turbulent flows under a neutrally-stratified condition in the LES model is valid. Here, L is the Obukhov length (m), g is the acceleration due to gravity $(\hbox {m s}^{-2})$, T is the air temperature ($\mathrm{K}$), $\overline{w'T'}$ is the sensible heat flux (${ \hbox {K m s}^{-1}}$), and $u_*$ is the friction velocity $({\hbox {m s}^{-1})}$. An overbar and prime denote a temporal average and fluctuation, respectively. A period with $|z/L| \le 0.05$ (Roth 2000) is regarded as fulfilling the neutrally-stratified condition.

By imposing the above conditions on the observational data, we obtained the following four 30-min periods: 0720–0750 LT (local time = UTC + 9 h) 22 January; 1650–1720 LT 30 January; 0740–0810 LT 2 February; and 1830–1900 LT 10 February, which are referred to as the D1, D2, D3, and D4 periods, respectively. The wind directions for each period calculated from the averaged horizontal velocity components are $49^\circ$, $358.8^\circ$, $353.8^\circ$, and $351.5^\circ$ for the D1–D4 periods, respectively.

To compare the LES results with the observations, it is necessary to use airflows observed at the Ujigawa Open Laboratory coming from the northern boundary of the analysis region of Kyoto passing through the analysis region, and not from the western or eastern boundaries. Because of the periodic conditions at the western and eastern boundaries, the flow through these lateral boundaries is unlikely to be accurately simulated by the LES model. This condition requires that wind directions be within a range of between approximately $355^\circ$ and $5^\circ$ based on the streamwise length and half the spanwise length of the analysis region [i.e., arctan (1 km/11 km)]. Overall, the wind directions in the periods D1–D4 are almost within the range of this condition, although those in the periods D3 and D4 are slightly shifted westwards from the condition. We confirmed that the area within at least 1 km westwards from the analysis region is dominated by land-use and building types similar to those in the analysis region. Thus, we concluded that the anemometer data taken during the four periods described above are appropriate for comparison with the LES results. However, the wind directions measured by the Doppler lidar deviate from those recorded by the sonic anemometer. The directions of the Doppler lidar in the D1 and D3 periods become more westerly with height, reaching $330^\circ$ at a height of 200 m, while those in the D2 and D4 periods are relatively constant with altitude and within a range between approximately $350^\circ$ and $0^\circ$. We discuss the possible influences of the variation of wind direction in Sect. 4.3. As explained above, none of the observed wind directions was oriented in a truly northerly fashion. Correspondingly, we rotated the streamwise directions to the mean of the wind directions measured by the sonic anemometer and the Doppler lidar.

4.3 Results

Figure 5a, b shows the time series of streamwise and spanwise velocity components produced by the LES model and measured by the sonic anemometer at a 25-m height, respectively. To avoid interference from the tower on the wind-speed profiles, the LES results are shown for a grid point 16 m north of the tower. It is seen that the LES turbulent fluctuations in both the streamwise and spanwise directions are comparable to those from the anemometer. Note that average spanwise velocity components are nearly zero, as indicated in Fig. 5b. The streamwise velocity component is greater in the D2 period than in the other periods. Comparison of the respective weather charts for the four time periods reveals the greater wind speeds in the D2 period to be caused by a large low-pressure system crossing the north-west Pacific Ocean off the coast of the Japanese Islands.

Figure 5c shows a comparison of the LES and observed vertical profiles of the mean streamwise velocity component. Both datasets are averaged over time, and the time-averaged LES data are averaged horizontally over a 16 m by 16 m area to the north of the tower to increase the representativeness of the simulated flows for the observation site. Note that, given the logarithmic scales used on both axes, the slopes of the mean streamwise velocity component in Fig. 5c suggest a power-law profile. According to Counihan (1975), the slopes of suburban and urban areas range between 0.21 and 0.28, making a power-law exponent of 1/4 suitable for reference, where it is seen that the slopes of the observations and the LES results are very similar to this value. We also examined the respective vertical profiles of the mean streamwise velocity component normalized by the mean streamwise velocity component at the 25-m height (see Online Resource 1, Fig. 2) and found that the LES and observed mean streamwise velocity components are quantitatively consistent. We conclude that this result provides good evidence for the reasonable performance of our LES model. In contrast, the slopes above approximately the 150-m height in the D1, D2, and D3 periods appear to deviate from the reference slope. In the case of the D1 and D3 periods, we assume this occurs because of the change in wind direction from northerly to westerly, as described in the previous sub-section. Another possible explanation for the deviation at the higher levels is that the stability conditions may not have been neutral at these heights during the observed periods. Because there were no observational data available to classify the stability condition above height of the sonic anemometer at 25 m, it is impossible to quantitatively reveal the stability above that height.

The vertical profiles of Reynolds stress in both the observations and the LES results are shown in Fig. 5d. Note that the Reynolds stress is normalized by the mean streamwise velocity components at each height. The Reynolds stress of the LES data is averaged horizontally over the same 16 m by 16 m area used for the mean streamwise velocity component. It is seen that the vertical profile of the LES data is within the range of differences found in the observation periods, which is a feature similar to that of the profiles normalized by the mean streamwise velocity component at the 25-m height (see Online Resource 1, Fig. 2). However, care is needed in comparing the LES results with the Doppler lidar data because the latter might include errors in representing perturbations of the wind speed as discussed below.

We now compare the results for the turbulence intensity, which is the ratio of the standard deviation of each velocity component $\sigma _i$ to the mean streamwise velocity component. As previously mentioned, the turbulence intensity was also averaged horizontally in the 16 m by 16 m area. Figure 6 compares the vertical profiles of turbulence intensity in the LES results and observations with the empirical form of the ESDU (1985), which is a database providing the turbulence characteristics of a neutrally-stratified atmospheric boundary layer based on various field measurements from around the world. In Fig. 6, all sonic-anemometer components fall within the rough-surface category given by the ESDU, which indicates suburban areas with $z_0$ between 0.1 and 0.5 m. Each component simulated by the LES model appears to capture the vertical distribution of that obtained by the ESDU within or around its upper and lower limits, at least below about the height of 150 m, while being slightly smaller than those of the sonic-anemometer observations. In fact, the values obtained from the sonic anemometer lie near the upper limit of the ESDU profile, suggesting that the LES results within the ESDU range are generally more favourable.

In contrast, there appears to be large discrepancies between the Doppler lidar observations and the LES results in terms of the u and v velocity components. The turbulence intensities for these components measured by the Doppler lidar are even larger than the upper limits of the ESDU, suggesting that the measurements may include an overestimating bias for the turbulence intensities. It is, in fact, commonly understood that the Doppler lidar measurements overestimate the turbulence intensities for the streamwise component. This characteristic was noted in Cañadillas et al. (2011), who showed that the results produced by the Doppler lidar are larger than those of sonic anemometers at various wind speeds and altitudes, and the deviations become larger with the decrease in wind speed. A close look at Figs. 5c and 6a indicates that the difference between Doppler lidar and the ESDU in terms of the streamwise turbulence intensity below 100 m decreases as the streamwise velocity conponent increases, in apparent confirmation of Cañadillas et al. (2011). For the Doppler lidar data above 100 m, changes in the wind direction and uncertainty in the stability, as revealed in the mean streamwise velocity component, may contribute to this overestimation. An overestimating tendency in the lidar data can also be found for the spanwise component, which has a mean value $\approx 0$.

The vertical component produced by the LES results appears to be consistent with both the lidar data and the ESDU profile, but the lidar tends to underestimate the vertical turbulence intensity, particularly in weaker wind-speed conditions.

Fig. 6

Vertical profiles of turbulence intensity from the observations, the LES model, and the empirical profiles provided by the ESDU (1985) for the a u, b v, and c w components. The dashed and dashed–dotted lines indicate the upper and the lower limits, respectively, of the rough-surface class based on the ESDU (1985)

Figure 7 shows the power spectra of the time series of each velocity component obtained from the LES results and the sonic anemometer at a height of 25 m. The spectra were calculated from the time series shown in Fig. 5, and the frequency f and velocity spectra E(f) are normalized to dimensionless form. The figure includes the empirical reference from Kaimal et al. (1972) derived from observations over a rural region. A close agreement is seen between the sonic anemometer and the reference results for all three components. The spectra from the sonic anemometer clearly represent an inertial subrange with a − 2/3 slope. Comparison of the LES spectra with the observations and empirical reference reveals that the spectra of the u and v components of the LES data are similar to those of the sonic anemometer data except in the highest frequency range. The lower frequency portion of the inertial subrange appears to be well reproduced for these components in the LES results.

However, the LES model is able to reproduce the vertical velocity components in only the lowest frequency portion of the inertial subrange, and it is possible that the grid spacing used in our modelling is insufficient for resolving the smallest eddies and their corresponding vertical motion. Further increases in the vertical resolution may be required to represent the small-scale vertical motion likely to be induced at the edges of buildings. However, we note that the spectral peak of the w component in the LES results agrees well with that of the sonic anemometer.

From the above comparisons, we conclude that the use of our LES model leads to a reasonable reproduction of the turbulent boundary-layer flow over actual buildings under a neutral stability condition, at least up to a height of about 150 m. We emphasize that, in general, the results produced by our LES model agree favourably with the observations within the range of differences among the chosen periods (D1–D4), even though our inflow condition employed an idealized turbulent flow generated in the driver region without realistic meteorological conditions. These results are sufficient here because our analysis of building-height variability focuses on altitudes below approximately 25 m (i.e., height $z=2.5h_{all}$), where the LES results show an especially close agreement with the observations, as shown in Figs. 6 and 7.

Fig. 7

Power spectra obtained from the sonic anemometer and the LES model at 25-m height plotted on logarithmic axes: a u, b v, and c w components. The dashed line indicates the empirical profile over a rural surface proposed by Kaimal et al. (1972)

Fig. 8

Vertical profiles of a time-averaged streamwise velocity component, b Reynolds stress, and c dispersive flux averaged spatially over the main region. These values are normalized by $U_{\infty }$. Red and blue lines denote the result of the CTL and UNI experiments, respectively. The vertical axis is normalized by $z = h_{all}$. Note that a logarithmic scale is used for the vertical axis

5 Sensitivity to Building-Height Variability

5.1 General Characteristics of Turbulent Flows

We now focus on the overall characteristics of turbulent flows in the CTL and UNI experiments, starting with the differences between the respective experiments.

Figure 8a, b shows the vertical profiles of the space- and time-averaged streamwise velocity component $\langle \overline{u}\rangle _{all}$ and Reynolds stress $-\langle \overline{u'w'}\rangle _{all}$ over the entire main region for the CTL and UNI experiments, respectively. Here, the angled brackets denote a spatial average, while the subscript all refers to the overall main region. Note that the values are normalized by the mean streamwise velocity component $U_{\infty }$ at the height of the boundary layer ($\delta$). The mean streamwise velocity components above height $z = h_{all}$ (i.e., above the canopy layer) are lower in the CTL experiment than in the UNI experiment, and in contrast, the velocity magnitudes below height $z = h_{all}$ for the CTL experiment are higher than in the UNI experiment. The Reynolds stress above height $z = h_{all}$ in the CTL experiment is larger than that in UNI experiment; furthermore, the level of peak Reynolds stress is higher in the CTL experiment than in the UNI experiment.

These differences between the CTL and UNI results can be attributed to the effects of building-height variability. Using the LES results of flows over idealized arrays of roughness blocks, Nakayama et al. (2011) showed that the mean velocity magnitude above the building height decreases with increasing building-height variability, and that the magnitude and height of the peak of the Reynolds stress both increase with building-height variability. Our results in terms of the streamwise velocity component and Reynolds stress are consistent with Nakayama et al. (2011).

Xie et al. (2008) carried out an LES investigation over block arrays with random and uniform heights, and found that both types of arrays produced similar turbulent kinetic energies below the average building height. The Reynolds stresses produced in the CTL and UNI experiments below height $z = h_{all}$ are consistent with their results. From Fig. 8b, it is seen that the Reynolds stress in the UNI experiment sharply increases around height $z = h_{all}$, which is likely caused by the presence of the uniform tops of buildings in the UNI experiment, resulting in increased wind shear and the generation of turbulence.

In Coceal et al. (2006), the velocity components $u_i$ were decomposed as

$$\begin{aligned} u_i=\langle \overline{u}_i \rangle + \overline{u}_{i}'' + u_{i}', \end{aligned}$$

(10)

where $\langle \overline{u}_i\rangle$ are the time- and space-averaged velocities, $\overline{u}_{i}''$ is the spatial variation of the time-averaged velocity, and $u_{i}'$ is the turbulent fluctuation. Coceal et al. (2006) showed that dispersive flux, which is defined as $\langle \overline{u}''\overline{w}''\rangle$, significantly contributes to the total momentum flux in the canopy layer in which the time-averaged velocities are spatially inhomogeneous. The vertical profiles of the dispersive flux normalized by $U_{\infty }$ in the CTL and UNI experiments are shown in Fig. 8c. Although the dispersive fluxes for both experiments have peaks just below $z = h_{all}$, the magnitude of the peak in the UNI experiment is larger than that in the CTL experiment. The UNI profile decreases sharply with height above the height of the peak. Above $z = h_{all}$, the dispersive flux in the CTL experiment is larger than that in the UNI experiment up to about $z = 3.5h_{all}$. Xie et al. (2008) performed an LES investigation to compare the dispersive flux in random and uniform block arrays. Their results suggest that both types of dispersive flux have peaks near the average building height, that the peaks obtained from uniform block arrays are stronger than those for random block arrays, and that the dispersive flux of uniform block arrays decreases much more abruptly with increasing height above the height of the peak than that of random block arrays. These characteristics are qualitatively consistent with our results. The dispersive fluxes in both the CTL and UNI experiments appear not to decrease linearly with height because the time-averaged velocities are not spatially homogeneous at heights above the canopy layer. Based on the results shown in Fig. 8a, b, we focus on the height $z=0.5h_{all}$ at which the difference between the CTL and UNI experiments is small, and the height $z=2.5h_{all}$ where clear differences are seen between the respective experiments.

Figure 9 shows the fields of time-averaged streamwise velocity component normalized by $U_{\infty }$ for the CTL and UNI experiments over an upstream region ($x = 1{-}5$ km) in which the business districts are located. The difference between the respective experimental results for the region appears to be small at $z = 0.5h_{all}$ except in areas along a major street around $y= 1.3$ km. This is likely caused by a stronger convergence of the streamwise velocity component on the street in the UNI experiment owing to enhancements arising from the presence of uniform-height buildings (i.e., in the UNI experiment, all lower building heights are raised to $z = h_{all}$). The velocity-deficit regions are reproduced at $z = 2.5h_{all}$ behind buildings in the CTL experiment, which contrasts to the smooth field of the time-averaged streamwise velocity component at $z = 2.5h_{all}$ in the UNI experiment.

Fig. 9

Horizontal cross-sections of the time-averaged streamwise velocity component normalized by $U_{\infty }$ in a the CTL experiment at $z = 0.5h_{all}$, b the UNI experiment at $z = 0.5h_{all}$, c the CTL experiment at $z = 2.5h_{all}$, and d the UNI experiment at $z = 2.5h_{all}$. An upstream part of the main region is shown. The legend indicating the wind speed is present to the right of each panel, while the grey shading indicates buildings

Figure 10 shows the fields of Reynolds stress normalized by $U_{\infty }$ for the CTL and UNI experiments over the upstream region. While the features are quite similar at $z = 0.5h_{all}$, the field at $z = 2.5h_{all}$ in the CTL results has larger values behind the buildings than in the UNI results, which indicates the important role of sparsely and randomly distributed buildings at and above $z = 2.5h_{all}$ in generating turbulence in the CTL experiment.

5.2 Analysis of Roughness Parameter

To quantitatively reveal the effects of building-height variability, we examined the relationships between the turbulent statistics and roughness parameters. The plan-area index $\lambda _{p}$ is used for this analysis because the CTL and UNI experiments have the same values for this parameter. Turbulent statistics were derived in each 1 km by 1 km area in a manner similar to that used to find the roughness parameters in Sect. 2.

Fig. 10

As Fig. 9, except with the corresponding Reynolds-stress results

5.2.1 Reynolds Stress

Figure 11 shows how the Reynolds stress normalized by $U_\infty$ in the CTL and UNI experiments changes as a function $\lambda _p$ at heights $z=0.5h_{all}$ and $z=2.5h_{all}$. The brackets with subscript 1 $\mathrm km^2$ indicate spatial averaging over a 1 km by 1 km area. The Reynolds stress at $z = 0.5h_{all}$ is very similar for the two experiments, which is consistent with the features shown in Fig. 10a, b. By contrast, the values at height $z = 2.5h_{all}$ in the CTL experiment increase with $\lambda _p$, while those in the UNI experiment are nearly independent of $\lambda _p$. In addition, the differences between the CTL and UNI results at $z = 2.5h_{all}$ are more apparent when $\lambda _p > 0.32$.

Fig. 11

Variations of Reynolds stress normalized by $U_{\infty }$ with $\lambda _p$ at heights a $z=0.5h_{all}$, and b $z=2.5h_{all}$

Fig. 12

Variations of $\lambda _{p}$ calculated a at $z = 0.5h_{all}$ ($\lambda _{{p}, 0.5h_{all}}$), b at $z = 2.5h_{all}$ ($\lambda _{{p}, 2.5h_{all}}$), and c $\lambda _{f}$ calculated at $z = h_{all}$ ($\lambda _{{f}, h_{all}}$) with $\lambda _{{p}}$ at the surface. Note that the scale of the vertical axis differs by panel

As shown in Figs. 9 and 10, the difference between the CTL and UNI experiments in terms of building distributions at $z = 2.5h_{all}$ has a significant effect on the turbulent flow results. To interpret this difference, we calculated the respective plan-area indices $\lambda _{p}$ at this altitude; i.e., for each experiment, if the building height in a grid cell is below $z = 2.5h_{all}$, the grid cell is regarded as having no buildings. Figure 12a, b shows $\lambda _{p}$ at $z = 0.5h_{all}$ (denoted by $\lambda _{{p}, 0.5h_{all}}$) and $z = 2.5h_{all}$ ($\lambda _{{p}, 2.5h_{all}}$), respectively, plotted against $\lambda _{p}$ at the surface for both the CTL and UNI experiments. Note that, in the UNI experiment, the value of $\lambda _{{p}, 0.5h_{all}}$ is the same as that of $\lambda _{p}$ at the surface, and that $\lambda _{{p}, 2.5h_{all}}$ is zero for this experiment. The difference between the CTL and UNI experiments in terms of $\lambda _{{p}, 0.5h_{all}}$ is not very large, confirming the similarity of the respective Reynolds stresses at $z = 0.5h_{all}$ in Fig. 10. In the CTL experiment, $\lambda _{{p}, 2.5h_{all}}$ rapidly increases if $\lambda _{p}> 0.32$, which appears to be consistent with the Reynolds-stress feature in the CTL experiment at $z = 2.5h_{all}$ as seen in Fig. 10. Based on these results, we suggest that the Reynolds stress from the CTL experiment at $z = 2.5h_{all}$ becomes greater at $\lambda _{p} > 0.32$ because some building clusters are still present at $z = 2.5h_{all}$ in this experiment.

The frontal-area index $\lambda _{f}$ is another important parameter for describing the geometrical characteristics of urban areas, and here we examine the frontal area of buildings above the height $h_{all}$. Figure 12c shows $\lambda _{f}$ above $z = h_{all}$ ($\lambda _{{f}, h_{all}}$) plotted against $\lambda _{p}$ for the CTL experiment (the figure does not include the corresponding values for the UNI experiment owing to the absence of buildings at that altitude). It is seen that $\lambda _{{f}, h_{all}}$ increases with $\lambda _{p}$, and sharply increases when $\lambda _{p} > 0.32$. These features agree well with the characteristics determined above for $\lambda _{{p}, 2.5h_{all}}$ and the Reynolds stress. According to these results, the effects of building-height variability on the Reynolds stress increase with $\lambda _{p}$ when $\lambda _{p}$ is $> 0.32$, and are closely linked to the higher values of $\lambda _{{p}, 2.5h_{all}}$ and $\lambda _{{f}, h_{all}}$ at such values of $\lambda _{p}$.

Interestingly, Zaki et al. (2011) found that the drag coefficient $C_d$ in wind-tunnel experiments, which is relevant to the Reynolds stress, increases with $\lambda _{p}$ when $\lambda _{p} > 0.32$ in flows over block arrays with random heights. A similar feature can also be found for the Reynolds stress and $C_d$ in the LES investigation of Nakayama et al. (2011). According to Zaki et al. (2011), this is because taller buildings, which contribute largely to the total drag in a block array (Xie et al. 2008), tend to be sparsely distributed and, therefore, despite the increase in $\lambda _{p}$, the flow pattern does not enter a skimming-flow regime (Oke 1988). Based on these previous studies and our results, $\lambda _{p} \approx 0.3$ can be regarded as a threshold at which the effects of building-height variability on the turbulent flow become apparent in various cities.

5.2.2 Momentum Transfer According to a Quadrant Analysis

As described in Sect. 1, turbulent coherent structures over urban surfaces are related to the physical process of turbulent momentum transfer. A quadrant analysis is a useful method for identifying the characteristics of the momentum transfer associated with coherent structures, and has been used in numerous studies of wall turbulence (Wallace 2016). This method divides the Reynolds stress into four components based on the signs of $u'$ and $w'$: outwards interaction (quadrant 1, $u'>0, w'>0$); ejection (quadrant 2, $u'<0, w'>0$); inwards interaction (quadrant 3, $u'<0, w'<0$) and sweep (quadrant 4, $u'>0, w'<0$). Raupach (1981) introduced conditional averaging using the threshold H to investigate the contribution to the Reynolds stress from the ith quadrant as

$$\begin{aligned} \langle u'w' \rangle _{i, H}= \lim _{T \rightarrow \infty } \frac{1}{T} \int _0^T u'(t)w'(t)I_{i, H}[u'(t), w'(t)] \mathrm{d}t, \end{aligned}$$

(11)

where the trigger indicator $I_{i, H}$ is defined as

$$\begin{aligned} I_{i, H}(u', w') = \left\{ \begin{array}{ll} 1, &{}\quad \hbox {if } (u', w') \text { is in quadrant } i \text { and if } |u'w'| \ge H| \overline{u'w'}|, \\ 0, &{}\quad \hbox {otherwise.} \\ \end{array} \right. \end{aligned}$$

(12)

The fraction of stress exceeding the threshold, which indicates the relative quantity of the ith quadrant, is

$$\begin{aligned} S_{i, H} = \langle u'w'\rangle _{i, H}/ \overline{u'w'}. \end{aligned}$$

(13)

where we note that the relationship

$$\begin{aligned} S_{1, 0}+S_{2, 0}+S_{3, 0}+S_{4, 0} =1 \end{aligned}$$

(14)

holds only for $H=0$. When the Reynolds stress is negative (as is normally seen in the boundary layer), $S_{2, 0}$ and $S_{4, 0}$ are positive, while $S_{1, 0}$ and $S_{3, 0}$ are negative.

Ejections and sweeps contribute to the downwards momentum flux, and are considered to be associated with organized turbulent motions as indicated in Sect. 1. Thus, the magnitude of ejections and sweeps is a good indicator for determining the characteristics of turbulent flows. To further reveal the relative roles of ejections and sweeps in vertical momentum transfer, we introduce the two parameters

$$\begin{aligned} \Delta S_0= & {} S_{4, 0} - S_{2, 0}, \end{aligned}$$

(15)

$$\begin{aligned} Ex= & {} (S_{1, 0} + S_{3, 0})/(S_{2, 0} + S_{4, 0}), \end{aligned}$$

(16)

where $\Delta S_0$ is the difference between sweeps and ejections, and Ex, which is called the exuberance (Shaw et al. 1983), is the ratio of unorganized ($S_1$ and $S_3$) motions to organized ($S_2$ and $S_4$) motions. The exuberance indicates the efficiency of transfer for the vertical momentum flux. Christen et al. (2007) used these parameters to investigate vertical momentum exchange in an urban district and elucidated the roles of coherent structures in momentum transport.

Figure 13 shows the vertical profiles of $\Delta S_0$ and Ex for the CTL and UNI experiments, which are averaged temporally and spatially in a manner similar to the profiles in Fig. 8, where $\Delta S_0$ in the CTL experiment is generally larger than in the UNI experiment, except for heights around $h_{all}$. This feature of $\Delta S_0$ contrasts to the vertical profile of the Reynolds stress shown in Fig. 8b, which indicates that the Reynolds stress is nearly identical in the CTL and UNI experiments below height $z = 0.5h_{all}$. This suggests that, despite the similarities in the Reynolds stress seen in the two experiments, the building-height variability in the CTL experiment changes the ratio of ejections to sweeps within the building canopy layer. In the upper layer from $z = 2.5h_{all}$ to $z = 10h_{all}$, both $\Delta S_0$ and the Reynolds stress are larger in the CTL experiment than in the UNI experiment. We consider that the increased Reynolds stress in this upper layer in the CTL experiment is caused by a sweep-dominated vertical flux.

Fig. 13

Vertical profiles of a $\Delta S_0$ and b Ex averaged spatially over the main region. Red and blue lines denote the results of the CTL and UNI experiments, respectively. The vertical axis is normalized by $h_{all}$, noting that a logarithmic scale is used for the vertical axis

Figure 13b shows the value of Ex below $z = 2.5h_{all}$ in the CTL experiment to be smaller than that in the UNI experiment. Below $z = 0.5h_{all}$, the decrease in Ex appears to be more pronounced in the CTL experiment than in the UNI experiment even though the respective Reynolds stresses are similar, as shown in Fig. 8b. This indicates that the transfer efficiency of the vertical momentum flux in the canopy layer is reduced by building-height variability. In contrast, the values of Ex in the CTL and UNI experiments are very similar at altitudes above height $z = 2.5h_{all}$, indicating that the transfer efficiency of the momentum flux above these altitudes is similar for both experiments.

Fig. 14

Variations of $\Delta S_0$ with $\lambda _{p}$ at a $z = 0.5h_{all}$ and b $z = 2.5h_{all}$, Ex at c heights $z = 0.5h_{all}$ and d $z = 2.5h_{all}$, and $E_{20}$ at e $z = 0.5h_{all}$ and f $z = 2.5h_{all}$

Based on the differences between the vertical profiles shown in Fig. 13, we focus on the heights $z=0.5h_{all}$ and $z=2.5h_{all}$ to reveal the relationship between building-height variability and turbulent-flow characteristics. Figure 14a, b shows variations in $\Delta S_0$ against $\lambda _{p}$ in the CTL and UNI experiments at these two altitudes. At $z = 0.5h_{all}$, sweeps are dominant among the contributions to the Reynolds stress for both experiments, which is consistent with previous results showing a stronger contribution of sweeps to the total momentum flux than ejections near and below the tops of block arrays (Raupach 1981; Coceal et al. 2007a). From Fig. 13a, it is seen that the contribution of sweeps in the CTL experiment is larger than in the UNI experiment, and in contrast, the value of $\Delta S_0$ at $z = 0.5h_{all}$ appears to be independent of $\lambda _{p}$ in both experiments. However, at $z = 2.5h_{all}$, the value of $\Delta S_0$ in the CTL experiment increases with $\lambda _{p}$ when $\lambda _{p} > 0.32$, while in the UNI experiment, it is independent of $\lambda _{p}$. The increase in $\Delta S_0$ in the CTL experiment is consistent with the Reynolds-stress results shown in Fig. 11b, thus suggesting that sweeps contribute to the increase in Reynolds stress for $\lambda _{p} > 0.32$. Similar results were noted by Kanda (2006).

Figure 14c and d shows Ex plotted against $\lambda _{p}$ in the CTL and UNI experiments at $z = 0.5$ and $2.5h_{all}$, respectively, where the difference in the value of Ex at $z = 0.5h_{all}$ increases with $\lambda _{p}$, suggesting the dominance of unorganized structures as $\lambda _{p}$ increases. As shown in Fig. 13b, at $z = 2.5h_{all}$ the values of Ex in both experiments are practically independent of $\lambda _{p}$.

By setting H in Eq. 12 to a value $>0$, we evaluate the extent to which extreme instantaneous momentum fluxes contribute to the total Reynolds stress in a certain period. We define the percentage contribution to the Reynolds stress of a value of $u'w'$ larger than the Reynolds stress by a factor of H using

$$\begin{aligned} E_H \equiv \sum _{i=1}^4 S_{i, H} = \sum _{i=1}^4 \langle u'w'\rangle _{i, H}/\overline{u'w'}. \end{aligned}$$

(17)

Unlike in Raupach (1981), in which each component of the momentum flux was evaluated, all of the components in Eq. 17 are added to assess the total of the extreme momentum fluxes. We set $H = 20$ here to extract extreme values of $u'w'$, with qualitatively similar results also found with $H = 15$ and $H = 10$. Thus, $H = 20$ is assumed to be a representative value.

Figure 14e, f shows the variations of $E_{20}$ with $\lambda _{p}$ at heights $z = 0.5$ and $z=2.5h_{all}$, respectively, for both experiments. The results at $z = 0.5h_{all}$ reveal small differences between the CTL and UNI experiments and are, in general, larger than those at $z = 2.5h_{all}$. This indicates that the flow is highly turbulent at $z = 0.5h_{all}$, and that the extreme values of the momentum flux contribute more significantly to the total momentum flux at this altitude. However, as the magnitude of $u'w'$ itself is low at $z = 0.5h_{all}$, the effects of the fluctuation itself may not be very strong. It is seen that, at $z = 2.5h_{all}$, the value of $E_{20}$ in the CTL experiment increases with $\lambda _{p}$, but is independent of $\lambda _{p}$ in the UNI experiment. Moreover, the shape of the relationship between $E_{20}$ at $z = 2.5h_{all}$ and $\lambda _{p}$ in the CTL experiment appears to be quite similar to that between $\lambda _{{p}, 2.5h_{all}}$ and $\lambda _{p}$ shown in Fig. 12b. This suggests that increasing the number of buildings at $z = 2.5h_{all}$ generates highly turbulent flows at higher values of $\lambda _{p}$.

The increase in the contribution from extreme values of $u'w'$ to the Reynolds stress at $z = 2.5h_{all}$ in the CTL experiment occurs because the building-height variability in this experiment leads to a higher momentum flux at this altitude as clearly indicated in Fig. 15a, which shows the horizontal cross-section of $E_{20}$ over a 1 km by 1 km area within one of the business districts. It is seen that high values of $E_{20}$ appear in areas around randomly and sparsely distributed buildings. In contrast, areas with higher $E_{20}$ values also correspond to areas with a low Reynolds stress and small value of Ex (see Fig. 15b, c), which indicates the small contribution of the extreme momentum flux around buildings to the total momentum flux, and is not related to organized turbulent motion. From the features demonstrated in Figs. 14 and 15, it is seen that the turbulent-flow characteristics and contributions of extreme momentum fluxes are significantly influenced by the presence of buildings with significant height variability.

Fig. 15

Horizontal cross-section of a $E_{20}$, b Reynolds stress normalized by $U_{\infty }$, and c Ex at $z=2.5h_{all}$ over a 1 km by 1 km area within the business district in the CTL experiment. The grey shading indicates buildings

We have shown the qualitative consistency of the Reynolds stress and quadrant analysis results, if averaged both in time and space, with that over block arrays with variable height. In contrast, the inhomogeneous profiles of the turbulent-flow characteristics (Fig. 15) suggest that the local characteristics of the turbulent-flow over urban surfaces are significantly influenced by the inhomogeneity of actual urban buildings, and would not be expected to be similar to that over idealized block arrays.

6 Summary and Conclusions

A LES investigation of the turbulent flow over the city of Kyoto has been conducted to investigate the effects of building-height variability on turbulence in the lower part of the urban boundary layer. Digital surface model data have reproduced the actual buildings of Kyoto in the LES model.

We used roughness parameters such as $H_{ave}$, $\sigma _H$, $\lambda _{{p}}$, and $\lambda _{{f}}$ to evaluate the morphological characteristics of buildings, and compared these parameters with those derived for Tokyo and Nagoya as well as for North American and European cities. For $\lambda _{p} > 0.3$, the value of $\lambda _{f}$ for Kyoto is small compared with the empirical values for Tokyo and Nagoya, but similar to those obtained for European cities. The relationship between $H_{ave}$ and $\sigma _H$ in Kyoto agrees closely with the empirical profile, and from these comparisons, the building morphological characteristics of Kyoto indicate a dense distribution, and buildings with a range of heights.

We compared the LES results with observations of atmospheric turbulence obtained using a sonic anemometer and a Doppler lidar at the Ujigawa Open Laboratory, which is an area included in the main region of the LES model. For this comparison, certain periods were extracted from the total set of observations to meet the weather conditions assumed in the LES model. The model is used to reproduce the observed characteristics of turbulence up to a height of about 150 m.

We carried out two experiments: one modelling the actual buildings of Kyoto (CTL), and one (UNI) in which all building heights were set to the average building height $h_{all}$ in the main region of the city. We find small differences between the CTL and UNI experiments in terms of the mean streamwise velocity component and the Reynolds stress at height $z = 0.5h_{all}$, but large differences at $z = 2.5h_{all}$. The spatial fields of time-averaged streamwise velocity component and Reynolds stress produced in the CTL experiment indicate regions of reduced velocity and large Reynolds stress behind sparsely and randomly distributed buildings at $z = 2.5h_{all}$; this contrasts with the UNI results, in which these fields at $z = 2.5h_{all}$ are smooth. We investigated the relationships between turbulent statistics and $\lambda _{p}$ evaluated over 1 km by 1 km areas to reveal the differences between the CTL and UNI experiments. The Reynolds stress in the CTL experiment at $z = 2.5h_{all}$ is larger than that in the UNI experiment when $\lambda _{p} > 0.32$, while the Reynolds stress at $z = 0.5h_{all}$ is similar for both experiments. We suggest that the increase in the Reynolds stress at $z = 2.5h_{all}$ is caused by the presence of building clusters at $z = 2.5h_{all}$ in the CTL experiment, and that a value of $\lambda _{p}\approx 0.3$ is the threshold above which the effects of building-height variability become obvious over various urban surfaces.

A quadrant analysis was used to investigate the characteristics of turbulent coherent flows. Sweeps in the CTL experiment at $z = 2.5h_{all}$ are found to increase with $\lambda _{p}$ for $\lambda _{p} > 0.32$, which is similar to that seen in the Reynolds stress for $\lambda _{p} > 0.32$, suggesting the increase in Reynolds stress is due to the presence of sweeps. The vertical momentum flux in the CTL experiment is associated with less transfer efficiency than that in the UNI experiment at $z = 0.5h_{all}$, which indicates that the building-height variability in the CTL experiment reduces the transfer efficiency in the canopy layer.

The contributions of the extreme instantaneous momentum flux to the total Reynolds stress were also investigated, with the amount of extreme momentum flux in the CTL experiment at $z = 2.5h_{all}$ depending strongly on the presence of buildings at this altitude. Examination of horizontal cross-sections reveals that areas with extreme momentum fluxes are distributed around buildings. However, the transfer efficiency of the Reynolds stress and momentum flux are small in areas with an extreme momentum flux, implying its negligible contribution around buildings to the net Reynolds stress, as well as the lack of association with coherent turbulent motions. The relationships between turbulent coherent structures and building-height variability were investigated through the use of space- and time-averaged profiles. However, future research on turbulent coherent structures over urban surfaces should focus on instantaneous and local structures, such as vortex structures behind high, isolated buildings (Park et al. 2015), and flow patterns in block arrays associated with coherent structures above blocks (Inagaki et al. 2012).