# Spectral-Element Simulation of the Turbulent Flow in an Urban Environment

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## Abstract

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## 1. Introduction to Urban Flows

#### 1.1. Structure of the Flow above the Urban Canopy

#### 1.2. Influence of an Urban Obstacle

#### 1.3. Description of the Flow Inside the Urban Roughness Sublayer

## 2. Governing Equations and Numerical Simulations

#### Mesh Design

## 3. Analysis of the Well-Resolved LES

#### 3.1. Analysis of the Mean Flow

#### 3.2. Analysis of the Turbulent Fluctuations

## 4. Comparison with Experimental Data

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Instantaneous visualization of the flow under study showing vortical structures identified with the ${\lambda}_{2}$ method [40]. The shown isosurface is $-50$ (scaled with the freestream velocity and the obstacle height) and the structures are colored in terms of the streamwise velocity from (dark blue) $-1$ to (dark red) 1.8.

**Figure 2.**Slices of the spectral-element mesh: (

**top**) slice at $z=0$ and (

**bottom**) slice at $y=0.1$ (where the lengths are scaled with the building height). Note that the GLL points within elements are not shown.

**Figure 3.**(

**Top**) Mean streamwise velocity U, where colors range from (dark blue) $-0.4$ to (dark red) 1.1. (

**Bottom**) Mean wall-normal velocity W, with colors ranging from (dark blue) $-0.35$ to (dark red) 1. The $y=0$ plane is shown.

**Figure 4.**Mean streamwise velocity U on the (

**top**) $z=0.85$ and (

**bottom**) $z=0.25$ planes, where colors range from (dark blue) $-0.4$ to (dark red) 1.1.

**Figure 5.**(

**Top**) Streamwise normal Reynolds stress ${u}_{\mathrm{rms}}^{2}$, where colors range from (dark blue) 0 to (dark red) 0.2. (

**Bottom**) Square of the rms spanwise velocity fluctuations ${v}_{\mathrm{rms}}^{2}$, where colors range from (dark blue) 0 to (dark red) 0.25. The $y=0$ plane is shown.

**Figure 6.**(Top) Non-dimensional mean streamwise velocity $U/{U}_{H}$ and (bottom) mean wall-normal velocity $W/{U}_{H}$ at $y/{W}_{b}=0$. Data extracted from the experiments by Monnier et al. [9]. Note that S denotes street separation.

**Figure 7.**(Top) Square of the root-mean-squared velocity fluctuations (top) ${u}_{\mathrm{rms}}^{2}/{U}_{H}^{2}$ and (bottom) ${v}_{\mathrm{rms}}^{2}/{U}_{H}^{2}$ at $y/{W}_{b}=0$. Data extracted from the experiments by Monnier et al. [9]. Note that S denotes street separation.

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**MDPI and ACS Style**

Stuck, M.; Vidal, A.; Torres, P.; Nagib, H.M.; Wark, C.; Vinuesa, R. Spectral-Element Simulation of the Turbulent Flow in an Urban Environment. *Appl. Sci.* **2021**, *11*, 6472.
https://doi.org/10.3390/app11146472

**AMA Style**

Stuck M, Vidal A, Torres P, Nagib HM, Wark C, Vinuesa R. Spectral-Element Simulation of the Turbulent Flow in an Urban Environment. *Applied Sciences*. 2021; 11(14):6472.
https://doi.org/10.3390/app11146472

**Chicago/Turabian Style**

Stuck, Maxime, Alvaro Vidal, Pablo Torres, Hassan M. Nagib, Candace Wark, and Ricardo Vinuesa. 2021. "Spectral-Element Simulation of the Turbulent Flow in an Urban Environment" *Applied Sciences* 11, no. 14: 6472.
https://doi.org/10.3390/app11146472