Parametrization of Drag and Turbulence for Urban Neighbourhoods with Trees

Abstract

Urban canopy parametrizations designed to be coupled with mesoscale models must predict the integrated effect of urban obstacles on the flow at each height in the canopy. To assess these neighbourhood-scale effects, results of microscale simulations may be horizontally-averaged. Obstacle-resolving computational fluid dynamics (CFD) simulations of neutrally-stratified flow through canopies of blocks (buildings) with varying distributions and densities of porous media (tree foliage) are conducted, and the spatially-averaged impacts on the flow of these building-tree combinations are assessed. The accuracy with which a one-dimensional (column) model with a one-equation ($k$–$l$) turbulence scheme represents spatially-averaged CFD results is evaluated. Individual physical mechanisms by which trees and buildings affect flow in the column model are evaluated in terms of relative importance. For the treed urban configurations considered, effects of buildings and trees may be considered independently. Building drag coefficients and length scale effects need not be altered due to the presence of tree foliage; therefore, parametrization of spatially-averaged flow through urban neighbourhoods with trees is greatly simplified. The new parametrization includes only source and sink terms significant for the prediction of spatially-averaged flow profiles: momentum drag due to buildings and trees (and the associated wake production of turbulent kinetic energy), modification of length scales by buildings, and enhanced dissipation of turbulent kinetic energy due to the small scale of tree foliage elements. Coefficients for the Santiago and Martilli (Boundary-Layer Meteorol 137: 417–439, 2010) parametrization of building drag coefficients and length scales are revised. Inclusion of foliage terms from the new parametrization in addition to the Santiago and Martilli building terms reduces root-mean-square difference (RMSD) of the column model streamwise velocity component and turbulent kinetic energy relative to the CFD model by 89 % in the canopy and 71 % above the canopy on average for the highest leaf area density scenarios tested: $0.50\hbox { m}^{2}~\hbox {m}^{-3}$. RMSD values with the new parametrization are less than 20 % of mean layer magnitude for the streamwise velocity component within and above the canopy, and for above-canopy turbulent kinetic energy; RMSD values for within-canopy turbulent kinetic energy are negligible for most scenarios. The foliage-related portion of the new parametrization is required for scenarios with tree foliage of equal or greater height than the buildings, and for scenarios with foliage below roof height for building plan area densities less than approximately 0.25.

Appendix: Testing the Column Model with CFD Model Results

In this section the fidelity with which a 1-D column model with $k$–$l$ turbulence closure reproduces profiles of spatially-averaged flow as simulated by a 3-D CFD model with standard $k$–$\varepsilon$ closure, is assessed for all urban block scenarios (i.e., with and without tree foliage). The suite of urban configurations and methodology are described in Sect. 2.4. Configurations with building plan density $\uplambda _{P} = 0.25$ are first evaluated, and analysis is subsequently extended to other $\uplambda _{P}$ values.

1.1 Intermediate Building Plan Area Density

RMSD values for the profiles of the spatially-averaged streamwise velocity component $\left\langle {\bar{{u}}} \right\rangle$, spatially-averaged turbulent kinetic energy $\left\langle {\bar{{k}}} \right\rangle$, and spatially-averaged Reynolds stress $\left\langle {\overline{u^{\prime }w^{\prime }}} \right\rangle$, are shown for all scenarios in Fig. 8, where the spatial average is over the outdoor atmosphere only. RMSD values are normalized by the scaling wind velocity $(u_{\tau })$ for $\left\langle {\bar{{u}}} \right\rangle$, and by its square for $\left\langle {\bar{{k}}} \right\rangle$ and $\left\langle {\overline{u^{{\prime }}w^{{\prime }}}} \right\rangle$, and therefore represent the difference between the models normalized by flow forcing.

Fig. 8

RMSD values of the spatially-averaged streamwise velocity component (a), TKE (b), and Reynolds stress (c) between the column and CFD models for all scenarios with building density $\uplambda _{P} = 0.25$. RMSD values are for two atmostpheric layers: canopy ($0 < z\le H$) and above-canopy ($H < z \le 2H$). Leaf area density, from left to right for each foliage height, is 0.06, 0.13, 0.25, and 0.50 m$^{2}$ m$^{-3}$. The $y$-axis scale is magnified for $\left\langle {\overline{u^{{\prime }}w^{{\prime }}}} \right\rangle$. $u_{\tau } = 0.45~\hbox {m s}^{-1}$

The Reynolds stress is always well-reproduced by the column model (i.e., normalized $\textit{RMSD} < 0.05$; Fig. 8c), and hence profiles of $\left\langle {\overline{u^{{\prime }}w^{{\prime }}}} \right\rangle$ are not a focus of the subsequent analysis. RMSD for both $\left\langle {\bar{{u}}} \right\rangle$ and $\left\langle {\bar{{k}}} \right\rangle$ is substantially reduced when the tree foliage layer protrudes above the building tops (i.e., Tree4 and Tree5; Fig. 8a, b). This is primarily true for $\left\langle {\bar{{u}}} \right\rangle$ above the canopy and $\left\langle {\bar{{k}}} \right\rangle$ in the canopy. The major part of this effect is not simply reduction of the magnitude of $\left\langle {\bar{{u}}} \right\rangle$ and $\left\langle {\bar{{k}}} \right\rangle$, as RMSD values normalized by local averages of $\left\langle {\bar{{u}}} \right\rangle$ and $\left\langle {\bar{{k}}} \right\rangle$, respectively, remain substantially lower for Tree4 and Tree5 relative to the other scenarios (not shown). Furthermore, this is true for all tree foliage densities, suggesting that even small leaf area densities (e.g. $0.06~\hbox {m}^{2}~\hbox {m}^{-3}$) above the roof height can substantially reduce column model error relative to the no-tree case.

These tendencies are related to the impact of tree foliage on the dispersive transport of momentum in the canopy in the CFD model (primarily downwards), which is not represented in the column model. When $L_\mathrm{D} = 0.06~\hbox {m}^{2}~\hbox {m}^{-3}$ is added in the lower half of the building canopy (Tree1), the dispersive transport is virtually unchanged, whereas it decreases by a factor of $\approx$4 when this same layer is added above the canopy (Tree5). It decreases further by a factor of $\approx$3 as leaf area density increases to $0.50~\hbox {m}^{2}~\hbox {m}^{-3}$ for Tree5, whereas it only decreases by 25 % for the same increase of $L_\mathrm{D}$ in the Tree1 case. Hence, tree foliage deep in the canopy has little effect on the dispersive (‘subgrid’) flow, regardless of its density, whereas density is more important for tree foliage above the building canopy. Clearly, the height of the tree foliage relative to the building tops is a critical variable in the determination of the effects of both elements on the flow. Note that these results are contingent on uniformly distributed foliage, and foliage clumping at crown-neighbourhood scales would presumably diminish the dampening effect of foliage on dispersive motions.

Profiles of $\left\langle {\bar{{u}}} \right\rangle$ and $\left\langle {\bar{{k}}} \right\rangle$ in Fig. 9 confirm that agreement between the models is better with foliage above the canopy (Tree5), and for dense foliage in the upper part of the canopy (Tree3, $L_\mathrm{D} = 0.50~\hbox {m}^{2}~\hbox {m}^{-3}$) than for cases without trees or foliage deeper in the canopy, particularly for TKE in the canopy. Figure 9 also confirms that even small densities above the canopy (i.e., Tree5) strongly influence the profiles of mean flow. The column model has most difficulty representing TKE in the canopy for both the non-treed scenario, as in Santiago and Martilli (2010), and scenarios with foliage in the canopy (Fig. 9b). Trees introduce several additional processes/terms in the canopy, some of which are not included in the column model formulation but which directly affect TKE (e.g., terms in the $\varepsilon$-equation, Eq. 4).

Fig. 9

Profiles of the spatially-averaged streamwise velocity component (a) and TKE (b) from the CFD (symbols) and column (lines with corresponding colours) models, for foliage height scenarios Tree1, Tree3, and Tree5 (see domain visualizations above plots), each with low and high leaf area densities as indicated in the legend. Building density is $\uplambda _{P} = 0.25$. Results from both models for the foliage-free (No Trees) case are plotted in each panel

The column model is able to reproduce the CFD model profiles of the spatially-averaged streamwise velocity component and TKE, for scenarios with tree foliage, as well or better than for the cases without trees (i.e., those of Santiago and Martilli 2010). RMSD values are less than 0.7 $u_{\tau }$, or its square $u_{\tau }^{2}$, for $\left\langle {\bar{{u}}} \right\rangle$ and $\left\langle {\bar{{k}}} \right\rangle$, respectively, both in the canopy and above it (Fig. 8a, b). RMSD values fall to about 10 % of these forcing values for scenarios with tree foliage extending higher than the rooftops; a likely explanation is that foliage renders the flow less 3-D and more amenable to prediction in a 1-D framework. In other words, dense tree foliage that extends above the buildings reduces the importance of dispersive processes to a greater degree.

Median RMSD values over all simulations (with and without tree foliage) at $\uplambda _{P} = 0.25$ are 0.13 and 0.31 for $\left\langle {\bar{{u}}} \right\rangle$ in and above the canopy, respectively, and 0.25 and 0.23 for $\left\langle {\bar{{k}}} \right\rangle$ in and above the canopy, respectively. Overall, the 1-D column model performs similar to or better than a column model with the Santiago and Martilli (2010) building-only parametrization, relative to the CFD model for all scenarios, for $\uplambda _{P} = 0.25$.

1.2 Low and High Building Plan Area Densities

The column-CFD model comparison is extended to other built densities. Building plan area densities $(\uplambda _{P})$ of 0.00, 0.06, 0.11, and 0.44 are simulated with tree foliage height and density variation as in Sect. 3.2. Column-CFD model differences for these built densities closely resemble those at $\uplambda _{P} = 0.25$ for $\left\langle {\bar{{u}}} \right\rangle$, $\left\langle {\bar{{k}}} \right\rangle$ and $\left\langle {\overline{u^{{\prime }}w^{{\prime }}}} \right\rangle$ (not shown). As for $\uplambda _{P} = 0.25$, RMSD values are larger above the canopy for $\left\langle {\bar{{u}}} \right\rangle$ and in the canopy for $\left\langle {\bar{{k}}} \right\rangle$, but overall the spatially-averaged CFD model profiles are well-reproduced by the column model (Fig. 10). Trees consistently reduce RMSD. RMSD values at these other built densities are of similar magnitude to, or smaller than, that for $\uplambda _{P} = 0.25$ (not shown). Average RMSD is less than or equal to that reported by Santiago and Martilli (2010) for the scenarios without trees. Hence, it is concluded that the column model performs sufficiently well for all $\uplambda _{P}$.

Fig. 10

Profiles of the spatially-averaged streamwise velocity component (a) and TKE (b) from the CFD (symbols) and column (lines with corresponding colours) models, for three building densities $(\uplambda _{P})$, with foliage density $L_\mathrm{D} = 0.50~\hbox {m}^{2}~\hbox {m}^{-3}$ and varying foliage height. The “Forest” scenario has foliage for $0 \le z \le H/2$, and no buildings

1.3 Sensitivity to Parameter $C_{\varepsilon 5}$ in the CFD Model

The $C_{\mathrm{e}5}$ parameter in Eq. 4 determines the sink of dissipation rate $(\varepsilon )$ in the CFD model, and spatially-averaged flow results are quite sensitive to this parameter. There is evidence that lower values of this parameter (relative to the theoretical value of 1.26 computed based on Sanz (2003), used here as the default) may be more accurate, at least relative to select wind-tunnel measurements (see Web Supplement). Hence, select scenarios in Figs. 4 and 8 are reproduced with $C_{\varepsilon 5}$ = 1.00 and $C_{\varepsilon 5} = 1.10$ in the CFD model: Tree2, $L_\mathrm{D} = 0.50~\hbox {m}^{2}~\hbox {m}^{-3}$; Tree4, $L_\mathrm{D} = 0.50~\hbox {m}^{2}~\hbox {m}^{-3}$; Tree4, $L_\mathrm{D} = 0.06~\hbox {m}^{2}~\hbox {m}^{-3}$. The column model is also re-run for each case and again with each source term in Table 1 removed, with modified sectional drag coefficients and length scales output from the CFD model. The ability of the column model to reproduce the CFD model profiles of $\left\langle {\bar{{u}}} \right\rangle$, $\left\langle {\bar{{k}}} \right\rangle$, and $\left\langle {\overline{u{\prime }w^{\prime }}} \right\rangle$ is effectively identical. RMSD values between the column and CFD models changes by less than 0.08, 0.03, and 0.02, respectively. As such, we conclude that the correspondence of the column model with the CFD model is not significantly affected by the choice of $C_{\varepsilon 5}$ in the CFD model over the range $C_{\mathrm{e}5} =$ 1.00–126. Furthermore, the same terms identified in Sect. 3.3 are significant (not shown), and hence it is concluded that terms deemed important for inclusion in the new parametrization are not affected by the choice of $C_{\varepsilon 5}$ in the CFD model over the range $C_{\mathrm{e}5} =$ 1.00–126.