Large-Eddy Simulation of the Gust Index in an Urban Area Using the Lattice Boltzmann Method

Abstract

We used numerical simulations to investigate the general relationship between urban morphology and the intensity of wind gusts in built-up areas at the pedestrian level. The simulated urban boundary layer developed over a 19.2 km (length) $\times$ 4.8 km (width) $\times$ 1.0 km (height) simulation domain, with 2-m resolution in all directions, to explicitly resolve the detailed shapes of buildings and the flow at the pedestrian level. This complex computation was accomplished using the lattice Boltzmann method and by implementing a large-eddy simulation model. To generalize the results, a new parameter that expresses the intensity of gusts (the gust index, ${\tilde{U}}_{ max})$ was defined as the local maximum wind speed divided by the freestream velocity. In addition, this parameter was decomposed into the mean wind-speed ratio, ${\tilde{U}}$ and turbulent gust ratio, ${\tilde{U}}^{{\prime }}$ to evaluate the qualities of gusts. These parameters were useful for quantitatively comparing the gust intensities within urban canopies at different locations or even among different experiments. In addition, the entire horizontal domain was subdivided into homogeneous square patches, in which both the simulated gust parameters and the morphological characteristics of building geometries were averaged. This procedure masked the detailed structure of individual buildings but retained the bulk characteristics of the urban morphology. At the pedestrian level, the gust index decreased with increasing building cover. Compared to ${\tilde{U}}$, the quantity ${\tilde{U}}^{{\prime }}$ notably contributed to the index throughout the range of plan area index $(\lambda _p)$ values. The dependences of all normalized wind-speed ratios transiently changed at $\lambda _p =~0.28$. In cases where $\lambda _p < 0.28, {\tilde{U}}$ decreased with increasing $\lambda _p$, although ${\tilde{U}}^{{\prime }}$ was almost constant. In cases where $\lambda _p > 0.28, {\tilde{U}}$ was almost constant and ${\tilde{U}}^{{\prime }}$ decreased with increasing $\lambda _p$. This was explained by the change in flow regimes within the building canyon. At a higher elevation above the canopy layer, $\lambda _p$ becomes less relevant to normalized wind-speed ratios, and instead the aerodynamic roughness length became important.

Appendix: Comparison of the Present LBM–LES Model to a Previous Wind-Tunnel Experiment and the Navier–Stokes LES Model

Letzel et al. (2008) validated the parallelized large-eddy simulation model (PALM) by comparing PALM results with wind-tunnel results from Martinuzzi and Tropea (1993). This suggested that PALM accurately reproduces flow around a single cube in a wind tunnel and we followed this validation study in comparing our LBM-based model with PALM and the wind-tunnel experiment.

The airflow around an isolated cube (shown in Fig. 9) was modelled using both numerical methods, and had previously been conducted in a wind-tunnel experiment. Moreover, PALM is an established model for atmospheric boundary-layer studies of flat and real cities (e.g., Inagaki et al. 2012; Letzel et al. 2012; Kanda et al. 2013; Keck et al. 2014; Park et al. 2015).

Figure 10 shows the vertical distribution of the average wind-speed ratio between the main flow, U and the inflow, $U_b$ obtained from the simulation results. The profiles were plotted for several locations (i.e., before, in the middle and after the cube). A number of different grid resolutions, N, were applied to the simulation, and profiles were in good agreement with the experimental results with increasing grid resolution.

Fig. 10

Vertical profiles of the average wind speed, U, normalized to the bulk inflow cross-section average wind speed, $U_b$. Experimental data are represented by the dotted lines. Each profile represents the position mentioned in Fig. 9

The vertical (cube centre) and horizontal (bottom layer) cross-sections of the time-averaged streamline are shown in Fig. 11. In the vertical cross-section when $N = 32$, a swirling flow line near the bottom of the cube on the downwind side can be observed. Such a streamline was not reproduced under coarser spatial resolutions. However, the streamlines for the horizontal cross-section did not greatly differ, regardless of the domain resolution.

Fig. 11

The streamlines around the cube centre vertical cross-section and lowermost horizontal cross-sectional. N is the number of grids used to resolve the cube

Figure 12 shows a comparison of the vertical profile of $U/U_b$ computed with PALM and LBM–LES models. Different resolutions (i.e., N = 16, 32 and 64) were used in both simulations, with PALM and LBM–LES profiles fitting well when N = 64 for all locations. However, the domain resolution of the actual simulation conducted herein was fixed to 2 m (N = 32), which was homogeneous in all directions, to reduce the time and cost of the computation.

Fig. 12

Comparison of the PALM and LBM–LES vertical profiles of $U/U_b$. Domain resolutions of a 3 m $(N =16)$, b 1.5 m $(N =32)$ and c 0.75 m $(N =64)$ were homogeneous in all directions $(\mathrm{d}x = \mathrm{d}y = \mathrm{d}z)$