# On the Experimental, Numerical and Data-Driven Methods to Study Urban Flows

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## Abstract

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## 1. Motivation

## 2. Fundamentals of Urban Flows

## 3. Experimental Studies

#### 3.1. Open-Environment Experiments

#### 3.1.1. Full-Scale Measurements

#### 3.1.2. Reduced-Scale Measurements

#### 3.2. Closed-Environment Experiments

#### Closed-Environment Experiments with Particle Image Velocimetry

#### 3.3. Experiments Combining Various Techniques

## 4. Numerical Studies

## 5. Flow Structures and Data-Driven Methods

## 6. **Conclusions and** Outlook

**deterministic**flow dynamics, free of noise and small flow scales [162,163]. More details about the algorithm and the performance of the aforementioned methods is presented in the Appendix A, where also some other variants of POD and DMD are briefly introduced. Based on the understanding of two basic concepts, the spectral and spatial flow complexities, defined in Ref. [161] and adapted to the identification of complex flows in Ref. [163], both SPOD and HODMD are suitable tools for the analysis of urban flows. However, it is important to remark that turbulent flows are composed of both deterministic and stochastic motions, both defining the flow physics. The present methods only identify deterministic motion, which is related to the large-scale coherent structures present in the flow, while the small-scale flow structures, generally understood as low-energy structures, are omitted. Nevertheless, understanding the flow dynamics driving the large size, most energetic, coherent structures present in the flow, would shed light on the physical mechanisms in charge of the pollutant dispersion (as it was tested in other complex problems in [164,165,166]). Hence, the new future trends should focus on continuing to apply and develop highly efficient data-driven tools to identify the main flow patterns in urban flows, not only focusing on the aforementioned classical tools, but also exploring new algorithms of machine learning—for instance, those based on non-linear functions. Combining the fields of artificial intelligence and fluid dynamics could provide new models, with reduced degrees of freedom and with predictive capacities, that would also consider stochastic flow motions, providing additional insight into the main mechanisms related to urban pollution, but this field is still barely unexplored. The great success of artificial intelligence in several other different fields (i.e., banking, earthquake predictions and computer vision, to name a few), suggests the high potential of this new research line which should be investigated in the near future. Identifying the main flow patterns with high accuracy, and developing high-fidelity ROMs, will shed light into new mechanisms related to the flow behaviour, thermal effects and pollutant dispersion in this type of environment.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Summary of Modal Decompositions

#### Appendix A.1. Singular Value Decomposition and Proper Orthogonal Decomposition

#### Appendix A.2. Spectral Proper Orthogonal Decomposition

#### Appendix A.3. Dynamic Mode Decomposition and Higher-Order Dynamic Mode Decomposition

**Step 1: dimension reduction.**Applying truncated singular value decomposition (SVD) to the snapshot matrix ${\mathit{V}}_{1}^{K}$ yields:$${\mathit{V}}_{1}^{K}\simeq \mathit{W}\Sigma {\mathit{T}}^{\top}\equiv \mathit{W}{\widehat{\mathit{T}}}_{1}^{K},\phantom{\rule{1.em}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}{\widehat{\mathit{T}}}_{1}^{K}=\Sigma {\mathit{T}}^{\top}.$$The matrix ${\widehat{\mathit{T}}}_{1}^{K}$ is the dimension-reduced snapshot matrix. The number of SVD modes retained in this approximation N is defined as the spatial complexity. These modes are selected as in the SVD algorithm presented in Appendix A.1. A (tunable) tolerance ${\epsilon}_{1}$ estimates the standard SVD error, as described in Equation (A6).**Step 2: the DMD-d algorithm for the dimension-reduced snapshots.**The high-order Koopman assumption is applied to the reduced snapshot matrix, resulting in:$${\widehat{\mathit{V}}}_{d+1}^{K}\simeq {\mathit{R}}_{1}{\widehat{\mathit{V}}}_{1}^{K-d}+{\mathit{R}}_{2}{\widehat{\mathit{V}}}_{2}^{K-(d-1)}+\dots +{\mathit{R}}_{d}{\widehat{\mathit{V}}}_{d}^{K-1}.$$After some calculations, the several Koopman operators ${\mathit{R}}_{1},\cdots ,{\mathit{R}}_{K}$ are grouped into a single matrix, the eigenvalue problem of which provides the DMD modes, frequencies and growth rates that define the DMD expansion (A7). This expansion is sorted in decreasing order of the mode amplitudes and it is further truncated by eliminating the modes such that:$${a}_{m}/{a}_{1}<{\epsilon}_{2},$$

#### Appendix A.4. Some Variants of DMD Not Related to Time-Lagged Snapshots

- Sparsity-promoting DMD [187]. This method uses convex optimization techniques to identify a smaller set of important modes.
- Extended DMD [188]. This algorithm includes more basis functions in the standard DMD approximation, a fact that allows the method to retain more modes, thus enabling the description of more complex dynamical systems.
- Optimized DMD [189]. An optimization problem is solved to compute the DMD expansion (A7). Both DMD and HODMD are purely linear-algebra-driven approaches compared to optimized DMD. Furthermore, other authors [190] have developed other linear-algebra-oriented algorithms which treat separable nonlinear least-squares problems.
- DMD variants to treat noisy data. Several authors have put an effort on identifying and removing noise from the analysed data. Among other techniques it is important to mention the method by Dawson et al. [191], which characterizes the noise properties; the total least-squares DMD by Hemati et al. [192], which combines standard DMD with total least squares; and the method by Takeishi et al. [193], which combines standard DMD with a Bayesian formulation.
- Multi-resolution DMD (mrDMD) [194]. The main idea behind mrDMD is to separate the high- and low-frequency events which generally occur in complex flows. The algorithm divides the snapshot matrix (A1) into several segments to identify high- and low-frequency modes depending on the segment length. The resulting DMD expansion (A7) is then represented by the various sub-expansions of DMD modes with different frequency ranges. A similar idea is behind the multi-resolution POD algorithm [173].

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**Figure 2.**Simplified urban model, indicating the geometrical parameters and the definition of the angle of incidence (AOI). Instantaneous vortical structures identified with the ${\lambda}_{2}$ method [22] are shown with an isosurface of $-40$ (scaled in terms of the freestream velocity ${U}_{\infty}$ and h). The structures are colored by streamwise velocity, ranging from (dark blue) $-1.2$ to (dark red) 1.8. Light grey indicates the bottom wall, and dark grey the buildings.

**Figure 4.**Instantaneous visualizations of the flow around two wall-mounted obstacles, showing vortical structures identified with the ${\lambda}_{2}$ method [22], with an isosurface of $-40$ (scaled in terms of ${U}_{\infty}$ and h). The flow exhibits three regimes depending on the obstacle separation, according to the classification by Oke [18]: (

**top**) skimming flow, (

**middle**) wake interference and (

**bottom**) isolated roughness. The structures are colored by streamwise velocity, ranging from (dark blue) $-1.2$ to (dark red) $1.8$.

**Figure 6.**Instantaneous flow visualizations showing vortical structures [22] around a wall-mounted square cylinder, with (

**left**) turbulent- and (

**right**) laminar-inflow conditions. We show ${\lambda}_{2}$ isosurfaces of $-40$ (based on ${U}_{\infty}$ and ${w}_{b}$), and the structures are colored by streamwise velocity, ranging from (dark blue) $-1.2$ to (dark red) $1.8$. Figures produced from the databse by Vinuesa et al. [41].

**Figure 7.**Instantaneous visualization showing the velocity magnitude in a rural-to-urban transition case. (

**a**) Neutral- and (

**b**) stable-stratification cases are shown, with the velocity magnitude normalized by ${U}_{\infty}$. The centerplane is projected on plane A’, and the horizontal plane is located at $z/h=0.1$. Reprinted from Ref. [108], with permission of the publisher (Springer Nature).

**Figure 8.**Illustration of the main vortical structures around a surface-mounted obstacle, i.e., the horseshoe vortex (

**A**), the roof vortex (

**B**), the vortices on the obstacle side (

**C**) and the arch vortex (

**D**). Adapted from Ref. [128], with permission of the publisher (Cambridge University Press).

**Figure 9.**Oil-film visualizations of the vortical structures around a wall-mounted obstacle with $b/h=2$ and ${w}_{b}/h=0.29$. The following incidence angles are shown: (

**a**) $\mathrm{AOI}={15}^{\circ}$, (

**b**) $\mathrm{AOI}={30}^{\circ}$, (

**c**) $\mathrm{AOI}={45}^{\circ}$ and (

**d**) $\mathrm{AOI}={60}^{\circ}$. Reprinted from Ref. [132], with permission of the publisher (Elsevier).

**Figure 10.**(

**Left**) Isosurfaces of (red) ${\Gamma}_{1}=0.4$, (blue) streamwise and (purple) spanwise root-mean-squared (rms) fluctuations, showing thresholds equal to 75% of the value indicated on each panel, for $\mathrm{AOI}={0}^{\circ}$. (

**Right**) Same as (

**left**), with ${\Gamma}_{1}=0.35$ and $\mathrm{AOI}={30}^{\circ}$. Reprinted from Ref. [87], with permission of the publisher (Springer Nature).

Studies | DNS | LES | RANS | Other |
---|---|---|---|---|

Vinuesa et al. [41], Coceal et al. [101,102], | ✓ | |||

Brandford et al. [103] and Lee et al. [106] | ||||

Belcher et al. [97] and Goulart et al. [117] | ✓ | Analytical models | ||

Michioka et al. [80], Boppana et al. [81], | ||||

Sullivan et al. [85], Nakayama et al. [88,89], | ||||

Kataoka and Mizuno [91], Giometto et al. [92], | ✓ | |||

Crylls et al. [96], Cheng and Porté-Agel [107], | ||||

Tomas et al. [108,110] and Eisma et al. [111] | ||||

Santiago et al. [83] and Dejoan et al. [86] | ✓ | $k-\epsilon $ [123] | ||

Inagaki et al. [95] | ✓ | Lattice-Boltzmann method [124] | ||

Sini et al. [19] | $k-\epsilon $ [123] | |||

Jacob and Sagaut [84] | Lattice-Boltzmann method [124] | |||

Tong et al. [82] | Momentum and energy balances | |||

Theurer et al. [98] and Davidson et al. [99] | Gaussian plume models | |||

Soulhac et al. [114] | SIRANE model [113] | |||

Hamlyn et al. [115] | Network models |

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**MDPI and ACS Style**

Torres, P.; Le Clainche, S.; Vinuesa, R.
On the Experimental, Numerical and Data-Driven Methods to Study Urban Flows. *Energies* **2021**, *14*, 1310.
https://doi.org/10.3390/en14051310

**AMA Style**

Torres P, Le Clainche S, Vinuesa R.
On the Experimental, Numerical and Data-Driven Methods to Study Urban Flows. *Energies*. 2021; 14(5):1310.
https://doi.org/10.3390/en14051310

**Chicago/Turabian Style**

Torres, Pablo, Soledad Le Clainche, and Ricardo Vinuesa.
2021. "On the Experimental, Numerical and Data-Driven Methods to Study Urban Flows" *Energies* 14, no. 5: 1310.
https://doi.org/10.3390/en14051310