Parametric Study of Urban-Like Topographic Statistical Moments Relevant to a Priori Modelling of Bulk Aerodynamic Parameters

Abstract

For a horizontally homogeneous, neutrally stratified atmospheric boundary layer (ABL), aerodynamic roughness length, $z_0$, is the effective elevation at which the streamwise component of mean velocity is zero. A priori prediction of $z_0$ based on topographic attributes remains an open line of inquiry in planetary boundary-layer research. Urban topographies – the topic of this study – exhibit spatial heterogeneities associated with variability of building height, width, and proximity with adjacent buildings; such variability renders a priori, prognostic $z_0$ models appealing. Here, large-eddy simulation (LES) has been used in an extensive parametric study to characterize the ABL response (and $z_0$) to a range of synthetic, urban-like topographies wherein statistical moments of the topography have been systematically varied. Using LES results, we determined the hierarchical influence of topographic moments relevant to setting $z_0$. We demonstrate that standard deviation and skewness are important, while kurtosis is negligible. This finding is reconciled with a model recently proposed by Flack and Schultz (J Fluids Eng 132:041203-1–041203-10, 2010), who demonstrate that $z_0$ can be modelled with standard deviation and skewness, and two empirical coefficients (one for each moment). We find that the empirical coefficient related to skewness is not constant, but exhibits a dependence on standard deviation over certain ranges. For idealized, quasi-uniform cubic topographies and for complex, fully random urban-like topographies, we demonstrate strong performance of the generalized Flack and Schultz model against contemporary roughness correlations.

Appendix: Topography Construction

The topographies summarized in Tables 3 and 4 were built in MatLab$^{\mathrm {TM}}$. Construction of the topographies involved designation of two topographic “stencils”, where block heights are either fixed to zero or are allowed to vary at pre-defined locations. For cases 1–10, a topographic stencil with 256 uniform-height, square-based blocks was distributed across the $N_x = N_y = 128$ computational grid (i.e., each block had side length, $L_x/16$; see also Fig. 1a). The height of the blocks was varied as $0.02 \le h_o/H \le 0.25$, and thus the standard deviation, $\sigma _h/H$, was changed systematically while the skewness and kurtosis were set to fixed values (recall the description of topographic moments in Table 2). This approach is appealing for the present purposes since it allows higher-order moments to remain fixed. For cases 11–26, a topographic stencil with 64 uniform height, square-based blocks were distributed across the $N_x = N_y = 128$ computational grid (i.e., each block had side length, $L_x/8$; see also Fig. 1b, c). The block heights of groups of blocks were fixed to one of four equivalent values ($h_{o1}, h_{o2}, h_{o3}, h_{o4}$), and these values were systematically varied. These block heights were varied randomly to attain the desired realistic variability of $s_k$ (cases 11–18) and $k_u$ (cases 19–26). For cases 27–46, the cases 1–10 topographic stencil was reused, but the heights of the blocks were varied in pair to manipulate the statistical moments. For example, for cases 39–42, the case 9 topography was used as the base, and then the height of half the blocks was increased while the height of the remaining half was decreased. By introducing this heterogeneity, the skewness was varied systematically. For case 47 to 56, all block heights (within the topographic stencil) were selected with MatLab$\mathrm {TM}$’s random number generator (see Fig. 1d).