Effects of Time-Dependent Inflow Perturbations on Turbulent Flow in a Street Canyon

Abstract

Urban flow and turbulence are driven by atmospheric flows with larger horizontal scales. Since building-resolving computational fluid dynamics models typically employ steady Dirichlet boundary conditions or forcing, the accuracy of numerical simulations may be limited by the neglect of perturbations. We investigate the sensitivity of flow within a unit-aspect-ratio street canyon to time-dependent perturbations near the inflow boundary. Using large-eddy simulation, time-periodic perturbations to the streamwise velocity component are incorporated via the nudging technique. Spatial averages of pointwise differences between unperturbed and perturbed velocity fields (i.e., the error kinetic energy) show a clear dependence on the perturbation period, though spatial structures are largely insensitive to the time-dependent forcing. The response of the error kinetic energy is maximized for perturbation periods comparable to the time scale of the mean canyon circulation. Frequency spectra indicate that this behaviour arises from a resonance between the inflow forcing and the mean motion around closed streamlines. The robustness of the results is confirmed using perturbations derived from measurements of roof-level wind speed.

Appendices

Appendix 1: Statistical Performance Measures

Given matching pairs of observations, $O_i$, and predictions, $P_i$, (i.e. measurements and LES values at the same location), statistical performance measures may be defined following Chang and Hanna (2004) and Santiago et al. (2007),

$$\begin{aligned} NMSE&= \frac{\sum \nolimits _{i=1}^{N}\left( O_i-P_i\right) ^2}{\sum \nolimits _{i=1 }^{N} O_iP_i} , \end{aligned}$$

(27a)

$$\begin{aligned} FB&= \frac{\sum \nolimits _{i=1}^{N}O_i - \sum \nolimits _{i=1}^{N}P_i}{\frac{1}{2}(\hat{O}+ \hat{P})} , \end{aligned}$$

(27b)

$$\begin{aligned} R&= \frac{\sum \nolimits _{i=1}^{N}\left( O_i-\hat{O}\right) \left( P_i-\hat{P}\right) }{\left[ \sum \nolimits _{i=1}^{N}(O_i-\hat{O})^2\right] ^{1/2}\left[ \sum \nolimits _{i=1}^{N}(P_i-\hat{P})^2\right] ^{1/2}} , \end{aligned}$$

(27c)

$$\begin{aligned} FAC 2&= \frac{1}{N}\sum \limits _{i=1}^{N}\chi _i\ ,\ \text {where~} \chi _i= {\left\{ \begin{array}{ll} 1,\ \text {for~}\ 0.5\le P_i/O_i \le 2.0,\\ 0,\ \text {otherwise,} \end{array}\right. } \end{aligned}$$

(27d)

where N is the total number of predictions or observations.⁶ The averages are given by

$$\begin{aligned} \hat{O}&= \frac{1}{N}\sum \limits _{i=1}^{N}O_i, \end{aligned}$$

(28a)

$$\begin{aligned} \quad \hat{P}&= \frac{1}{N}\sum \limits _{i=1}^{N}P_i. \end{aligned}$$

(28b)

The normalized mean square error ($NMSE$), correlation coefficient (R) and fractional bias ($FB$) are standard, while $FAC 2$ represents the fraction of predicted values that lie within a factor of two of the observations. The hit rate,

$$\begin{aligned} q =&\frac{1}{N}\sum \limits _{i=1}^{N}\chi _i\ ,\ \text {where~} {\chi }_i= {\left\{ \begin{array}{ll} 1,\ \text {for~}\ \left| \frac{O_i-P_i}{O_i}\right| \le \varDelta _r\ \text {or~} |O_i-P_i|\le \varDelta _a, \\ 0,\ \text {otherwise,} \end{array}\right. } \end{aligned}$$

(29)

is closely related to $FAC 2$, and measures the probability of detection (Wilks 2011); $\varDelta _r$ is the allowed relative deviation and $\varDelta _a$ is the allowed absolute deviation. The values of $\varDelta _r=0.25$ and $\varDelta _a=0.05$ follow the validation studies of Eichhorn (2004) and Santiago et al. (2007).

Appendix 2: Frequency Spectrum of a Mesoscale Simulation

In order to confirm the plausibility of perturbation periods $T\in [{25}\,{\hbox {s}},{1200}\,{\hbox {s}}]$, high-resolution output from the WRF mesoscale model was analyzed. Five nested domains covering Hong Kong with $280\times 220$, $151\times 163$, $160\times 181$, $181\times 202$ and $220\times 220$ gridpoints and resolutions of 30, 10, 3.3, 1.1 and 0.37 km were considered, with a single-layer urban canopy model used for the finest domain. The model was initialized using initial and lateral boundary conditions at $1^{\circ }\times 1^{\circ }$ from the National Centers for Environmental Prediction.

Here we analyze a single case for which the simulation was initiated at 0600 UTC (1400 LST) 9 January 2017 and integrated until 1200 UTC (2000 LST) 11 January 2017. During the simulation period, the study area was under the influence of the southernmost periphery of a winter anticyclone. A 6-h time series was extracted from a point inside the Mong Kok neighbourhood of Hong Kong, with the corresponding frequency spectrum shown in Fig. 15. Peaks at 125, 200 and $620\,\mathrm{s}$ can be seen.

Fig. 15

Frequency spectrum from the WRF model simulation. The data were extracted from the lowest vertical level (approximately $50\,\mathrm{m}$) at $(22.32215^{\circ }, 114.16994^{\circ })$

Appendix 3: Scaling of the Co-WIN Data

Let u denote the time series of wind speed from the Community Weather Information Network (see Sect. 6). A normalized variable is defined as

$$\begin{aligned} \tilde{u} = \frac{u-\overline{u}}{\sigma ^\prime }, \end{aligned}$$

(30)

where the mean $\overline{u}$ and standard deviation $\sigma ^\prime$ are given by

$$\begin{aligned} \overline{u}&= \frac{1}{N} \sum _{i=1}^N u(t_i), \end{aligned}$$

(31a)

$$\begin{aligned} \sigma ^\prime&= \left[ \frac{1}{N} \sum _{i=1}^N ( u(t_i)-\overline{u})^2 \right] ^\frac{1}{2}. \end{aligned}$$

(31b)

By construction $\tilde{u}$ has a mean of zero and a standard deviation of 1. In Sect. 6, $\tilde{u}$ is used in place of u to define the observations, i.e.

$$\begin{aligned} X_o = U_{LS}\big [1 + \tilde{u}(t)\big ], \end{aligned}$$

(32)

which is a natural means of accommodating differences in the magnitudes of $U_{LS}$ and u.