Validation of Simplified Urban-Canopy Aerodynamic Parametrizations Using a Numerical Simulation of an Actual Downtown Area

Abstract

A steady-state Reynolds-averaged Navier–Stoke computational fluid dynamics (CFD) investigation of boundary-layer flow over a major portion of downtown Abu Dhabi is conducted. The results are used to derive the shear stress and characterize the logarithmic region for eight sub-domains, where the sub-domains overlap and are overlaid in the streamwise direction. They are characterized by a high frontal area index initially, which decreases significantly beyond the fifth sub-domain. The plan area index is relatively stable throughout the domain. For each sub-domain, the estimated local roughness length and displacement height derived from CFD results are compared to prevalent empirical formulations. We further validate and tune a mixing-length model proposed by Coceal and Belcher (Q J R Meteorol Soc 130:1349–1372, 2004). Finally, the in-canopy wind-speed attenuation is analysed as a function of fetch. It is shown that, while there is some room for improvement in Macdonald’s empirical formulations (Boundary-Layer Meteorol 97:25–45, 2000), Coceal and Belcher’s mixing model in combination with the resolution method of Di Sabatino et al. (Boundary-Layer Meteorol 127:131–151, 2008) can provide a robust estimation of the average wind speed in the logarithmic region. Within the roughness sublayer, a properly parametrized Cionco exponential model is shown to be quite accurate.

Appendix

We compare our estimated roughness length and displacement height values to those prescribed by ten prevailing formulations found across the literature, four of which incorporate roughness-element height variability.

Originally derived for vegetated surfaces, the Ra94 method provides reasonably accurate results in urban environments (Bottema and Mestayer 1998; Grimmond and Oke 1999),

$$\begin{aligned} d_{Ra94}= & {} \left[ {1+\left( {\frac{\hbox {exp}\left[ {-\left( {C_{dl} 2\lambda _F } \right) ^{0.5}} \right] -1}{\left( {C_{dl} 2\lambda _F } \right) ^{0.5}}} \right) } \right] h, \end{aligned}$$

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$$\begin{aligned} z_{0Ra94}= & {} \left[ {\left( {1-\frac{d_{Ra} }{h}} \right) \hbox {exp}\left( {-k\frac{U(h)}{{u_*} }+\psi _h } \right) } \right] h, \end{aligned}$$

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where

$$\begin{aligned} \frac{{u_*} }{U(h)}=\min \left[ {\left( {C_S +C_{Dv} \lambda _F } \right) ^{0.5},\left( {\frac{{u_*} }{U(h)}} \right) _{\mathrm{max}} } \right] . \end{aligned}$$

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Here, $C_S$ and $C_{Dv}$ are the drag coefficients for the substrate surface at height h in the absence of roughness elements, and of an isolated roughness element mounted on the surface, respectively, $C_{dl}$ is a free parameter, $\psi _h$ is the RSL influence function and U(h) and ${u_*}$ are the wind speed at the average roof height and the friction velocity, respectively. The values specified by Raupach (1994) are: $C_S =0.003, C_{Dv} =0.3, \psi _h =0.193, C_{dl} =7.5$ and $\left( {{u_*} /U(h)} \right) _{\mathrm{max}} =0.3$. Although these constants vary depending on the nature of the roughness elements (Bottema and Mestayer 1998), a sensitivity analysis is beyond the scope of the present study.

The Bo98 method is a simplification of Bottema (1995, 1997) for use in urban areas. The method assumes that all of the drag experienced by the flow is due to substrate roughness and includes a mutual sheltering parameter,

$$\begin{aligned} d_{Bo98}= & {} \lambda _P ^{0.6}h, \end{aligned}$$

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$$\begin{aligned} z_{0Bo98}= & {} \left( {h_V -d_{Bo98} } \right) \hbox {exp}\left( {-\frac{k}{\left( {0.5 C_{dh} \lambda _F } \right) ^{0.5}}} \right) h, \end{aligned}$$

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where $C_{dh}$ is the drag coefficient of buildings. Instead of the prescribed value of 0.8 for an isolated obstacle, we use the calculated value for each stripe.

The empirical formulations of Macdonald et al. (1998) include a drag correction coefficient, $\beta$, and a fitting constant, $\alpha$, that controls the increase of d / h with $\lambda _p$

$$\begin{aligned} d_{MC98}= & {} \left[ {1+\alpha ^{-\lambda _p }\left( {\lambda _p -1} \right) } \right] h, \end{aligned}$$

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$$\begin{aligned} z_{0MC98}= & {} \left[ {\left( {1-\frac{d_{MC98} }{h}} \right) \exp \left\{ {-\left[ {0.5\hbox { }\beta \hbox { }\frac{C_{dh} }{\kappa ^{2}}\left( {1-\frac{d_{MC98} }{h}} \right) } \right] ^{-0.5}\hbox { }} \right\} } \right] h, \end{aligned}$$

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where Macdonald et al. (1998) suggest $C_{dh} = 1.2$ and, based on wind-tunnel data from Hall et al. (1996), $\alpha = 4.43$, and $\beta = 1$ for staggered arrays, and $\alpha = 3.59$, and $\beta = 0.55$ for square arrays. The former coefficients are used, along with the calculated value of $C_{dh}$ for each stripe.

The simplest method, often called the “rule-of-thumb” relates the average building height to $z_0$ and d through a coefficient. Both GO98 and CC02 methods suggest values of $d_{GO98} =0.7h, z_{0GO98} =0.1h$ and $d_{CC02} =0.83h, z_{0CC02} =0.053h$. It is unclear how accurate all the above-mentioned relations are when applied to actual towns and cities, as there has been no study to test their validity for non-cubic arrays (Padhra 2010).

Four morphometric methods that incorporate variable roughness-element heights are tested: MH11, MH13, Ka13 and Zh16. The first method divides the urban canopy into layers and calculates a cumulative-height normalized d and drag balance, based on uniform arrays and an attenuation coefficient proposed by Macdonald (2000). Kent et al. (2017) argue that this process is overly complex and thus a relation based on the standard deviation of roughness-element heights has been developed (Millward-Hopkins et al. 2013)

$$\begin{aligned} d_{MH13}= & {} \left[ {\frac{f_{d,MH13} }{h}+\left( {\left( {0.2375\ln \left( {\lambda _p } \right) +1.1738} \right) \frac{\sigma _h }{h}} \right) } \right] h, \end{aligned}$$

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$$\begin{aligned} z_{0MH13}= & {} \left[ {\frac{f_{0,MH13} }{h}+\left( {\hbox {exp}\left( {0.8867\lambda _f } \right) -1} \right) \left( {\frac{\sigma _h }{h}} \right) ^{\mathrm {exp}\left( {2.3271\lambda _f } \right) }} \right] h, \end{aligned}$$

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where,

$$\begin{aligned} f_{d,MH13}= & {} \left\{ {{\begin{array}{ll} \frac{19.2\lambda _p -1+\hbox {exp}\left( {-19.2\lambda _p } \right) }{19.2\lambda _p \left[ {1-\hbox {exp}\left( {\lambda _p } \right) } \right] }&{} \quad \hbox {for}\quad \lambda _p \ge 0.19\\ \frac{117\lambda _p +\left( {187.2\lambda _p ^{3}-6.1} \right) \left[ {1-\hbox {exp}\left( {-19.2\lambda _p } \right) } \right] }{\left( {1+114\lambda _p +187\lambda _p ^{3}} \right) \left[ {1-\hbox {exp}\left( {-19.2\lambda _p } \right) } \right] }&{} \quad \hbox {for}\quad \lambda _p <0.19 \\ \end{array} }} \right. . \end{aligned}$$

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$$\begin{aligned} f_{0,MH13}= & {} \left( {1-\frac{d_{MH13} }{h}} \right) \hbox {exp}\left[ {-\left( {0.5C_{dh} \kappa ^{-2}\lambda _p } \right) ^{-0.5}} \right] h, \end{aligned}$$

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However, with height-dependent morphological indexes, we are able to re-write the relations originally proposed by Millward-Hopkins et al. (2011) such that

$$\begin{aligned} d_{MH11}= & {} \int _0^{h_{\mathrm{max}} } f_{d,MH11} \left( {\lambda _p (z)} \right) \mathrm{d}z, \end{aligned}$$

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$$\begin{aligned} z_{0MH11}= & {} \left( {1-\frac{d_{MH11} }{h}} \right) \hbox {exp}\left[ {-\left( {0.5C_{dh} \kappa ^{-2}\lambda _p (0)} \right) ^{-0.5}} \right] h, \end{aligned}$$

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where,

$$\begin{aligned} f_{d,MH11} \left( {\lambda _p (z)} \right) =\left\{ {{\begin{array}{ll} \frac{19.2\lambda _p (z)-1+\hbox {exp}\left( {-19.2\lambda _p (z)} \right) }{19.2\lambda _p (z)\left[ {1-\hbox {exp}\left( {\lambda _p (z)} \right) } \right] }h&{} \quad \hbox {for}\quad \lambda _p (z)\ge 0.19 \\ \frac{117\lambda _p (z)+\left( {187.2\lambda _p (z)^{3}-6.1} \right) \left[ {1-\hbox {exp}\left( {-19.2\lambda _p (z)} \right) } \right] }{\left( {1+114\lambda _p (z)+187\lambda _p (z)^{3}} \right) \left[ {1-\hbox {exp}\left( {-19.2\lambda _p (z)} \right) } \right] }h&{} \quad \hbox {for}\quad \lambda _p (z)<0.19 \\ \end{array} }}\right. . \end{aligned}$$

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Once more, the calculated value of $C_{dh}$ for each stripe is used.

The Ka13 model is derived based on horizontally-averaged statistics of LES of the city of Tokyo and simple arrays from the literature. Aerodynamic parameters are determined through a least-square regression and an empirical model based on five geometric parameters. Kanda et al. (2013) argue that $h_{\mathrm{max}}$ is a more suitable scaling parameter than h as it is found to coincide with the upper limit of d, such that

$$\begin{aligned} d_{Ka13} =\left[ {c_0 X^{2}+\left( {a_0 \lambda _p^{b_0 } -c_0 } \right) X} \right] h_{\mathrm{max}}, \end{aligned}$$

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where $a_0 =1.29$, $b_0 =0.36$ and $c_0 =-0.17$. X is the representative building height above the average building height $\left( {\sigma _h +h} \right)$, relative to the maximum building height,

$$\begin{aligned} X=\frac{\sigma _h +h}{h_{\mathrm{max}} } \end{aligned}$$

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for $0\le X\le 1$. The roughness length in the Ka13 method is a modification to that of MC98,

$$\begin{aligned} z_{0Ka13} =\left( {b_1 Y^{2}+c_1 Y+a_1 } \right) z_{0MC98}, \end{aligned}$$

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where $Y=\frac{\lambda _p\sigma _h}{h}$ for $0\le Y$, and $a_1 =0.71$, $b_1 =20.21$ and $c_1 =-0.77$ are empirically derived coefficients. The impact of $\lambda _p$ and $\sigma _h$ on $z_0$ is accounted for by Y, which tends to zero for homogenous arrays.

Finally, the Zh16 model has been derived through LES of urban-like topographies wherein statistical moments of the topography have been systematically varied (Zhu et al. 2016). These authors find that there is a strong relationship between d and the standard deviation of building heights, considering ground area, such that

$$\begin{aligned} d_{Zh16} =\alpha _d \langle \sigma _h\rangle , \end{aligned}$$

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where $\alpha _d =1.69$. Similarly, they find strong dependencies between $z_0, \langle \sigma _h\rangle$ and $\langle s_k\rangle$, and propose a hybrid roughness correlation between the Flack and Schultz (2010) and Ito et al. (2011) models

$$\begin{aligned} z_{0Zh16} =\left\{ {{\begin{array}{ll} \alpha \langle \sigma _h\rangle \left( {1+\beta \langle s_k\rangle } \right) &{} \quad \hbox {for}\quad \langle \sigma _h\rangle /\langle h\rangle <1.15 \\ \alpha \langle \sigma _h\rangle \left( {1+\langle s_k\rangle } \right) ^{\beta } &{} \quad \hbox {for}\quad \langle \sigma _h\rangle /\langle h\rangle \ge 1.15 \\ \end{array} }} \right. , \end{aligned}$$

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where the model parameters, $\alpha \approx 0.1$ and $\beta =0.9$, have been tuned for atmospheric flows over urban-like topographies. Traditional geometric parameters that do not consider ground surface $(h, \sigma _h$ and $s_k)$ were also used to test the above model giving $d_{Zhu16}$ and $z_{0Zhu16}$.